**Part 2**

**Transportation and Communication**

116 Fuzzy Logic – Emerging Technologies and Applications

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Weiser; M.; Gold, R.; & Brown, J. S. (1999) *Ubiquitous computing* - Retrieved 9 December 2010 from http://www.research.ibm.com/journal/sj/384/weiser.html. Witter, G.P. (2005) *Metaciência e Psicologia*, (Portuguese language) ISBN: 8575161075 , São

Zadeh, L.A. (1965). "Fuzzy sets". Information and Control 8 (3): 338–353. doi:10.1016/S0019-

Zadeh, L.A. (1968). "Fuzzy algorithms". Information and Control 12 (2): 94–102.

Zemankova-Leech, M. (1983). Fuzzy Relational Data Bases. Ph. D. Dissertation. Florida State

Zimmermann, H. (2001). Fuzzy set theory and its applications. Boston: Kluwer Academic

*interaction issues for mobile computing in a variable work context*, Int. J. Human-

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Computer Studies 60 ,771–797.

Paulo, Brazil. Editora: ALINEA

9958(65)90241-X. ISSN 0019-9958.

Publishers. ISBN 0-7923-7435-5.

doi:10.1016/S0019-9958(68)90211-8. ISSN 0019-9958.

IEEE.

Group.

University.

Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms (E.P. Klement

IFSA World Congress and 20th NAFIPS International Conference, pp. 1407-1412,

**7** 

*USA* 

 *Taiwan (R.O.C.)* 

**Fuzzy-Logic Analysis of the** 

 **in Atmospheric Turbulence** 

*2Department of Aviation Mechanical Engineering, China University of Science and Technology,* 

C. Edward Lan1 and Ray C. Chang2

**FDR Data of a Transport Aircraft** 

*1 Department of Aerospace Engineering, University of Kansas, KS,* 

The aerodynamics of a jet transport in severe atmospheric turbulence, in particular involving plunging motion, is complex in that unsteady aerodynamic effects are significant and not well known. For instance, the aircraft response may lag behind the change in angle of attack and/or control surface deflections. Because of the change in angle of attack, the wing vortex wake may be pulsating. Coupled with the aircraft motion, the pulsating vortex wake would significantly affect the tail aerodynamics and hence, the aircraft stability and control characteristics. These are just a few possible phenomena in aircraft response to be identified. Unfortunately, these aerodynamic characteristics cannot be identified with existing ground testing techniques. Therefore, at present the only option to estimate the aircraft aerodynamic characteristics in severe atmospheric turbulence is to analyze the data from Flight Data Recorders (FDR). Traditional methods of system identification in aerodynamics, such as the maximum likelihood method (MMLE) (Maine & Iliff, 1986), the least-square or the stepwise regression method (Klein, 1981), or the Extended Kalman Filter (EKF) (Minkler & Minkler, 1993; Gelb 1982), have not been demonstrated to be applicable to estimating the unsteady aerodynamics based on these FDR data. Therefore, a more robust model identification technique would be needed. In addition, the established aerodynamic models should be directly usable in flight simulation. To satisfy these goals, the Fuzzy Logic Modeling (FLM) technique is adopted in the present application. The technique used here has been applied to model identification of a fighter aircraft from flight test data (Wang, et al. 2001; Wang, et al. 2002); aerodynamic estimation of transport aircraft from Flight Data Recorder (FDR) data (Lan & Guan 2005; Weng, et al. 2008; Chang, et al. 2009); identification of uncommanded motions from wind-tunnel dynamic free-to-roll test data (Lan, et al., Jan. 2008; Lan, et al., May 2008); and non-aerodynamic problems with the FDR data (Lee & Lan

In the following, the present fuzzy logic algorithm will be described in some detail. It follows with some simple verification examples in Section 3. In Section 4, application of the

**1. Introduction** 

2003; Lan, et al. 2006), just to name a few.

C. Edward Lan1 and Ray C. Chang2

*1 Department of Aerospace Engineering, University of Kansas, KS, 2Department of Aviation Mechanical Engineering, China University of Science and Technology, USA Taiwan (R.O.C.)* 

#### **1. Introduction**

The aerodynamics of a jet transport in severe atmospheric turbulence, in particular involving plunging motion, is complex in that unsteady aerodynamic effects are significant and not well known. For instance, the aircraft response may lag behind the change in angle of attack and/or control surface deflections. Because of the change in angle of attack, the wing vortex wake may be pulsating. Coupled with the aircraft motion, the pulsating vortex wake would significantly affect the tail aerodynamics and hence, the aircraft stability and control characteristics. These are just a few possible phenomena in aircraft response to be identified. Unfortunately, these aerodynamic characteristics cannot be identified with existing ground testing techniques. Therefore, at present the only option to estimate the aircraft aerodynamic characteristics in severe atmospheric turbulence is to analyze the data from Flight Data Recorders (FDR). Traditional methods of system identification in aerodynamics, such as the maximum likelihood method (MMLE) (Maine & Iliff, 1986), the least-square or the stepwise regression method (Klein, 1981), or the Extended Kalman Filter (EKF) (Minkler & Minkler, 1993; Gelb 1982), have not been demonstrated to be applicable to estimating the unsteady aerodynamics based on these FDR data. Therefore, a more robust model identification technique would be needed. In addition, the established aerodynamic models should be directly usable in flight simulation. To satisfy these goals, the Fuzzy Logic Modeling (FLM) technique is adopted in the present application. The technique used here has been applied to model identification of a fighter aircraft from flight test data (Wang, et al. 2001; Wang, et al. 2002); aerodynamic estimation of transport aircraft from Flight Data Recorder (FDR) data (Lan & Guan 2005; Weng, et al. 2008; Chang, et al. 2009); identification of uncommanded motions from wind-tunnel dynamic free-to-roll test data (Lan, et al., Jan. 2008; Lan, et al., May 2008); and non-aerodynamic problems with the FDR data (Lee & Lan 2003; Lan, et al. 2006), just to name a few.

In the following, the present fuzzy logic algorithm will be described in some detail. It follows with some simple verification examples in Section 3. In Section 4, application of the

as the variables in the compatibility analysis and eventually forming the data for specific fuzzy models. In the present Chapter, *<sup>i</sup> y* is defined to be an estimated aerodynamic coefficient of force or moment, and *<sup>r</sup> x* are the variables of the input data. The numbers of the internal functions (i.e. cell's numbers) are quantified by the total number of membership

The values of each fuzzy variable, such as the angle of attack, are divided into several ranges, each of which represents a membership function with ( ) *A xr* as its membership grade. One membership function from each variable constitutes a fuzzy cell. For the *i*th cell, the corresponding membership grades are represented by ( ) *<sup>i</sup> Ar r x* , *r*=1, 2,…, *k*. In other words, the membership functions allow the membership grades of the internal functions for a given set of input variables to be assigned. For a given system with input variables 1 2 ,,,, *r k xx x x* of one data point, the recorded values of each input variables are normalized by using (*xr* - *xr,min*)/(*xr,max-xr,min*) to transform them into the ranges of [0, 1]. The range, (*xr,max - xr,min*), represents the scaling factor and usually is assumed to have a larger range than what actually appears in the data with numerous data points to be more generally applicable for the resulting model. In the present application in aerodynamics, it is empirically assumed to be 1.8 times larger. Generally, overlapped straight lines, triangles or

The membership functions partition the input space into many fuzzy subspaces, which are called the fuzzy cells. The total number of fuzzy cells is *nN N N N* 1 2 *r k* . For a variable *<sup>r</sup> x* , the number of membership function is *Nr* . Each fuzzy cell is in a different combination from others formed by taking one membership function from each input

Let *N* be the number of membership functions and *j* be the index for the *j*-th membership functions. Then the membership grades for triangular and parabolic shapes can be described

 *For j =3 to N- m, where m is equal to the greater number of 0 and integer of (N-2)/2:* 

parabolas are frequently the shapes used to represent the grades.

functions (see below).

variable.

as follows:

*1)N = 2:* 

2)N 3:

 *A(xr) = xr , j =1* 

*For j N* – *m*

 *A*(*xr*) = 1 – *xr* , *j* =2

 *A*(*xr*) = *xr*/*du*, 0  *xr du*

*A*(*xr*) = (1 – *xr*)/(1– *du*), *du xr* 1 where *du* = *x1*\*(*j* – 2), and *x1* =1.0/(*N* – *m* –*1*).

where *dd* = *x2*\*(*j* –*N* + *m*), and *x2* = 1.0/(*m*+1).

*A*(*xr*)=(*dd* – *xr*)/*dd*, 0 *xr dd A*(*xr*)= (*dd* – *xr*)/(*dd* –1), *dd xr* 1

**2.1.1 Triangular membership functions** 

FLM algorithm in aerodynamic model identification for a jet transport in severe atmospheric turbulence will be described in detail. Unsteady aerodynamics will be emphasized. Conclusions will follow in Section 5.

#### **2. Fuzzy logic modeling**

The general idea of the FLM technique is to set up the relations between system input and output variables. There are two approaches in the FLM technique. One is the fuzzy set approach, involving fuzzy sets, membership functions, weighting factors, and the if-then fuzzy rules (Zadeh 1973). The process involves three stages: fuzzification, fuzzy rule inference and defuzzification. The second approach is the internal function approach, involving the internal functions, membership functions, and the output cells (Takagi & Sugeno 1985). The same three stages mentioned above can also be identified. Since the first approach does not provide continuous derivatives needed in aerodynamics, the second approach will be utilized in the present paper.

Basically, the present FLM algorithm represents a multi-dimensional, nonlinear interpolation scheme without requiring explicit functional forms between the input and output variables. In application, complex motions or relations involving many variables can be treated. Conceptually, each motion variable is divided into a number of ranges in which values of the membership functions are assigned. Each combination of membership functions, one from each motion variable, constitutes a fuzzy cell. Each fuzzy cell contributes to the prediction of the value of outcome equal to its internal function with an associated weighting factor. The latter represents an assembly of the membership grades of all variables. The final prediction of outcome is equal to the weighted average of contributions of all fuzzy cells. This overview will be repeated later by way of equations or formulas.

Two main tasks are involved in the present FLM process. One is the identification of the coefficients of the internal functions. The other one is structure identification to identify the optimal structure of fuzzy cells of the model, in other words, the optimal number of membership functions for each variable. Details of fuzzification, fuzzy rule inference and defuzzification stages in the present FLM technique are described in the following (Wang, et al. 1998, 2001, 2002).

#### **2.1 Fuzzification**

In this stage, many internal functions are defined to cover the ranges of the influencing variables (i.e. input variables). The ranges of the input variables are all transformed into the domain of [0,1]. The membership grading also ranges from 0 to 1.0, with "0" meaning no effect from the corresponding internal function, and "1" meaning a full effect. These internal functions are assumed to be linear functions of input variables as follows:

$$P^{i} = y\_i(\mathbf{x}\_1, \mathbf{x}\_2, \cdots, \mathbf{x}\_r, \cdots \mathbf{x}\_k) \ = p\_0^i + p\_1^i \mathbf{x}\_1 + \cdots + p\_r^i \mathbf{x}\_r + \cdots \cdot p\_k^i \mathbf{x}\_k \tag{2.1}$$

where *<sup>i</sup> <sup>r</sup> p* , *r*=0, 1, 2,…, *k*, are the coefficients of internal functions *<sup>i</sup> y* , and *k* is number of input variables; *i*=1, 2, …, *n*, and *n* is the total number of fuzzy cells.

The recorded data in FDR, such as flight altitude (*h*), calibrated airspeed (CAS), angle of attack (), accelerometer readings (*ax*, *ay*, and *az*), and Euler angles (, , and ), etc is chosen as the variables in the compatibility analysis and eventually forming the data for specific fuzzy models. In the present Chapter, *<sup>i</sup> y* is defined to be an estimated aerodynamic coefficient of force or moment, and *<sup>r</sup> x* are the variables of the input data. The numbers of the internal functions (i.e. cell's numbers) are quantified by the total number of membership functions (see below).

The values of each fuzzy variable, such as the angle of attack, are divided into several ranges, each of which represents a membership function with ( ) *A xr* as its membership grade. One membership function from each variable constitutes a fuzzy cell. For the *i*th cell, the corresponding membership grades are represented by ( ) *<sup>i</sup> Ar r x* , *r*=1, 2,…, *k*. In other words, the membership functions allow the membership grades of the internal functions for a given set of input variables to be assigned. For a given system with input variables 1 2 ,,,, *r k xx x x* of one data point, the recorded values of each input variables are normalized by using (*xr* - *xr,min*)/(*xr,max-xr,min*) to transform them into the ranges of [0, 1]. The range, (*xr,max - xr,min*), represents the scaling factor and usually is assumed to have a larger range than what actually appears in the data with numerous data points to be more generally applicable for the resulting model. In the present application in aerodynamics, it is empirically assumed to be 1.8 times larger. Generally, overlapped straight lines, triangles or parabolas are frequently the shapes used to represent the grades.

The membership functions partition the input space into many fuzzy subspaces, which are called the fuzzy cells. The total number of fuzzy cells is *nN N N N* 1 2 *r k* . For a variable *<sup>r</sup> x* , the number of membership function is *Nr* . Each fuzzy cell is in a different combination from others formed by taking one membership function from each input variable.

Let *N* be the number of membership functions and *j* be the index for the *j*-th membership functions. Then the membership grades for triangular and parabolic shapes can be described as follows:

#### **2.1.1 Triangular membership functions**

*1)N = 2: A(xr) = xr , j =1 A*(*xr*) = 1 – *xr* , *j* =2 2)N 3:

120 Fuzzy Logic – Emerging Technologies and Applications

FLM algorithm in aerodynamic model identification for a jet transport in severe atmospheric turbulence will be described in detail. Unsteady aerodynamics will be emphasized.

The general idea of the FLM technique is to set up the relations between system input and output variables. There are two approaches in the FLM technique. One is the fuzzy set approach, involving fuzzy sets, membership functions, weighting factors, and the if-then fuzzy rules (Zadeh 1973). The process involves three stages: fuzzification, fuzzy rule inference and defuzzification. The second approach is the internal function approach, involving the internal functions, membership functions, and the output cells (Takagi & Sugeno 1985). The same three stages mentioned above can also be identified. Since the first approach does not provide continuous derivatives needed in aerodynamics, the second

Basically, the present FLM algorithm represents a multi-dimensional, nonlinear interpolation scheme without requiring explicit functional forms between the input and output variables. In application, complex motions or relations involving many variables can be treated. Conceptually, each motion variable is divided into a number of ranges in which values of the membership functions are assigned. Each combination of membership functions, one from each motion variable, constitutes a fuzzy cell. Each fuzzy cell contributes to the prediction of the value of outcome equal to its internal function with an associated weighting factor. The latter represents an assembly of the membership grades of all variables. The final prediction of outcome is equal to the weighted average of contributions of all fuzzy cells. This overview will

Two main tasks are involved in the present FLM process. One is the identification of the coefficients of the internal functions. The other one is structure identification to identify the optimal structure of fuzzy cells of the model, in other words, the optimal number of membership functions for each variable. Details of fuzzification, fuzzy rule inference and defuzzification stages in the present FLM technique are described in the following (Wang, et

In this stage, many internal functions are defined to cover the ranges of the influencing variables (i.e. input variables). The ranges of the input variables are all transformed into the domain of [0,1]. The membership grading also ranges from 0 to 1.0, with "0" meaning no effect from the corresponding internal function, and "1" meaning a full effect. These internal

1 2 0 11 ( , , , , ) *i i <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>P</sup> i rk <sup>r</sup> r k <sup>k</sup> <sup>y</sup> xx x x p p <sup>x</sup> <sup>p</sup> <sup>x</sup> <sup>p</sup> <sup>x</sup>* (2.1)

, , and 

), etc is chosen

*<sup>r</sup> p* , *r*=0, 1, 2,…, *k*, are the coefficients of internal functions *<sup>i</sup> y* , and *k* is number of input

The recorded data in FDR, such as flight altitude (*h*), calibrated airspeed (CAS), angle of

functions are assumed to be linear functions of input variables as follows:

), accelerometer readings (*ax*, *ay*, and *az*), and Euler angles (

variables; *i*=1, 2, …, *n*, and *n* is the total number of fuzzy cells.

Conclusions will follow in Section 5.

approach will be utilized in the present paper.

be repeated later by way of equations or formulas.

al. 1998, 2001, 2002).

**2.1 Fuzzification** 

where *<sup>i</sup>*

attack (

**2. Fuzzy logic modeling** 

 *For j =3 to N- m, where m is equal to the greater number of 0 and integer of (N-2)/2:* 

 *A*(*xr*) = *xr*/*du*, 0  *xr du A*(*xr*) = (1 – *xr*)/(1– *du*), *du xr* 1 where *du* = *x1*\*(*j* – 2), and *x1* =1.0/(*N* – *m* –*1*). *For j N* – *m A*(*xr*)=(*dd* – *xr*)/*dd*, 0 *xr dd A*(*xr*)= (*dd* – *xr*)/(*dd* –1), *dd xr* 1 where *dd* = *x2*\*(*j* –*N* + *m*), and *x2* = 1.0/(*m*+1).

functions is not performed in estimating derivatives. In the present application, aerodynamic derivatives are all estimated with a central difference scheme, which will be presented later. Because overlapped triangular membership function is simple and involves less computing time, it is the method to represent the grades of membership functions in the present FLM technique. Comparison of computed results based on these two types of

**0**

**0**

**0.2**

**0.4**

**0.6**

**A(x)**

A fuzzy cell is formed by taking one membership function from each variable, as indicated earlier. The total number of cells is the number of possible combinations by taking one membership function from each input variable. For every cell, it has a fuzzy rule to guide the input and output relations. For the jth data point, the rule of the *i*th cell is stated (Wang, et

*if* 1, *<sup>j</sup> x is* 1 1, ( ) *<sup>i</sup> A x , and if j* 2, *<sup>j</sup> x is* 2 2, ( ) *<sup>i</sup> A x , ... and if j <sup>k</sup>*, *<sup>j</sup> x is* , ( ) *<sup>i</sup> A x for the j k k <sup>j</sup> th data point, then the cell* 

1, 2, , , 0 1 1, , , ( , ,, , ) *i i <sup>i</sup> <sup>i</sup> <sup>i</sup> Px x x x j j <sup>r</sup> <sup>j</sup> <sup>k</sup> j j r r <sup>j</sup> k k <sup>j</sup> p p x p x p x* (2.2)

**0.8**

**1**

**0.2**

**0.4**

**0.6**

**A(x)**

**0.8**

**1**

**0 0.2 0.4 0.6 0.8 1 x N=3**

**<sup>1</sup> <sup>2</sup> <sup>3</sup>**

**4**

**0 0.2 0.4 0.6 0.8 1 x N=5**

**5**

membership function will be illustrated later.

**0**

**1**

**0**

Fig. 2. Parabolic membership functions

*output is equal to its internal function:* 

**2.3 Fuzzy rule inference** 

al. 1998) as:

**0.2**

**0.4**

**0.6**

**A(x)**

**0.8**

**0.2**

**0.4**

**0.6**

**A(x)**

**0.8**

**1**

**0 0.2 0.4 0.6 0.8 1 x N=2**

**0 0.2 0.4 0.6 0.8 1 x N=4**

#### **2.1.2 Parabolic membership functions**

*1)N=2:* 

$$A(\mathbf{x}\_r) = \mathbf{x}\_{r\_r} \quad j = 1$$

$$A(\mathbf{x}\_r) = 1 - \mathbf{x}\_{r\_r}j = 2$$

*2)N 3:* 

 *For j = 3 to N-m, where m is again equal to the greater number of 0 and the integer of (N-2)/2:* 

$$A(\mathbf{x}\_{l}) = -\mathbf{x}\_{r}^{2} \;/\, d\_{u}^{2} + 2\mathbf{x}\_{r} \;/\, d\_{u}\,, 0 \le \mathbf{x}\_{r} \le d\_{u}$$

$$A(\mathbf{x}\_{l}) = -(\mathbf{x}\_{r}^{2} - 2d\_{u}\mathbf{x}\_{r} + 2d\_{u} - 1) \;/\, (1 - 2d\_{u} + d\_{u}^{2}), d\_{u} \le \mathbf{x}\_{r} \le 1$$

$$\text{where } d\_{u} = \Delta \mathbf{x}\_{l} \text{\*}(j - 2), \text{ and } \Delta \mathbf{x}\_{l} = \mathbf{1}.0 \!/\!/\!(\text{N} - m - 1).$$

$$j \ge \text{N} - m$$

$$A(\mathbf{x}\_{l}) = \mathbf{x}\_{r}^{2} \;/\, d\_{d}^{2} - 2\mathbf{x}\_{r} \;/\, d\_{d} + \mathbf{1}.0 \!/\:/ \mathbf{0} \le \mathbf{x}\_{r} \le d\_{d}$$

$$A(\mathbf{x}\_{l}) = \left(\mathbf{x}\_{r}^{2} - 2d\_{d}\mathbf{x}\_{r} + d\_{d}^{2}\right) / \left(\mathbf{1} - 2d\_{d} + d\_{d}^{2}\right), d\_{d} \le \mathbf{x}\_{r} \le \mathbf{1}$$

$$\text{where } d = \pm 1.0^{6} \text{ (M} \cdot \text{m} \cdot \text{s}) \text{ and } \mathbf{x}\_{r} = 1.0^{6} \text{ (m} \cdot \text{m} \cdot \text{s})$$

*where dd = x2\*(j –N + m), and x2 = 1.0/(m+1).* 

The membership functions are illustrated in Fig. 1 for triangular shapes and Fig. 2 for parabolic shapes. In Fig. 1, although the membership functions are continuous functions, there are discontinuities in slopes at some points. However, differentiation of membership

Fig. 1. Triangular membership functions

functions is not performed in estimating derivatives. In the present application, aerodynamic derivatives are all estimated with a central difference scheme, which will be presented later. Because overlapped triangular membership function is simple and involves less computing time, it is the method to represent the grades of membership functions in the present FLM technique. Comparison of computed results based on these two types of membership function will be illustrated later.

Fig. 2. Parabolic membership functions

#### **2.3 Fuzzy rule inference**

122 Fuzzy Logic – Emerging Technologies and Applications

 *For j = 3 to N-m, where m is again equal to the greater number of 0 and the integer of (N-*

**2.1.2 Parabolic membership functions** 

 *A(xr) =* 2 2 / 2 / ,0 *r u ru r u xd xd xd* 

*x1\*(j – 2), and* 

 *A(xr) =* 2 2 / 2 / 1.0,0 *r d rd r d xd xd xd*

*x2\*(j –N + m), and* 

 *A(xr) =* 2 2 ( 2 2 1) /(1 2 ), 1 *r ur u u uu r x dx d d d d x*

*x1 =1.0/(N – m –1).* 

*x2 = 1.0/(m+1).* 

The membership functions are illustrated in Fig. 1 for triangular shapes and Fig. 2 for parabolic shapes. In Fig. 1, although the membership functions are continuous functions, there are discontinuities in slopes at some points. However, differentiation of membership

**A(x)**

**A(x)**

**0 0.2 0.4 0.6 0.8 1**

**0 0.2 0.4 0.6 0.8 1** **0 0.2 0.4 0.6 0.8 1 x N=3**

> **1 2 3 4**

**0 0.2 0.4 0.6 0.8 1 x N=5**

**5**

 *A(xr) =* 22 2 ( 2 ) /(1 2 ), 1 *r dr d d dd r x dx d d d d x*

**0 0.2 0.4 0.6 0.8 1 x N=2**

**0 0.2 0.4 0.6 0.8 1 x N=4**

 *A(xr) = xr , j =1* 

 *A*(*xr*) = 1 – *xr* , *j* =2

**A(x)**

**A(x)**

**0 0.2 0.4 0.6 0.8 1**

**0 0.2 0.4 0.6 0.8 1**

Fig. 1. Triangular membership functions

*1)N=2:* 

*2)N 3:* 

*2)/2:* 

 *where du =* 

 *j > N – m* 

*where dd =* 

A fuzzy cell is formed by taking one membership function from each variable, as indicated earlier. The total number of cells is the number of possible combinations by taking one membership function from each input variable. For every cell, it has a fuzzy rule to guide the input and output relations. For the jth data point, the rule of the *i*th cell is stated (Wang, et al. 1998) as:

*if* 1, *<sup>j</sup> x is* 1 1, ( ) *<sup>i</sup> A x , and if j* 2, *<sup>j</sup> x is* 2 2, ( ) *<sup>i</sup> A x , ... and if j <sup>k</sup>*, *<sup>j</sup> x is* , ( ) *<sup>i</sup> A x for the j k k <sup>j</sup> th data point, then the cell output is equal to its internal function:* 

$$P^i(\mathbf{x}\_{1,j}, \mathbf{x}\_{2,j}, \dots, \mathbf{x}\_{r,j}, \dots \mathbf{x}\_{k,j}) = p\_0^i + p\_1^i \mathbf{x}\_{1,j} + \dots + p\_r^i \mathbf{x}\_{r,j} + \dots + p\_k^i \mathbf{x}\_{k,j} \tag{2.2}$$

( ) *i i rt rt r i*

*p* 

<sup>ˆ</sup> ( ,..., , ,..., ) ( ) 2( ) <sup>ˆ</sup> *<sup>m</sup> i n*

*i i j j k k j r j*

*<sup>r</sup>* is the convergence factor or the step size in the gradient method; subscript index *t*

*SSE y x x p p*

ˆ [ ( ),..., ( )]

*y product A x A x x <sup>p</sup> product A x A x*

denotes the iteration sequence, and *<sup>r</sup>*, *<sup>j</sup> x* =1.0 if r=0 in Eq. (2.6b). Usually, the magnitude of

1

1

The iteration during the search sequence stops when one of the following three criteria is

1) Cost= *SSEt* <sup>1</sup>

*SSE SSE SSE*

3) max *t t* (2.9)

In the above criteria, Cost= *SSEt* is the sum of squared errors (SSE) in current iteration to be denoted by "Cost" and RER=(cost\_current - cost\_previous)/cost\_current (i.e. the relative

specified maximum iteration number. The convergence of modeling is achieved only when

 and <sup>2</sup> 

*s*

*k k j r j i i rt rt r j j <sup>n</sup> s s*

*k kj i i t t jj <sup>n</sup> s s*

*s*

[ ( ), , ( )] 2( ) <sup>ˆ</sup>

[ ( ), , ( )] 2( ) <sup>ˆ</sup>

<sup>2</sup> *t t t*

*<sup>r</sup>* is chosen based on that of the gradient. Eq. (2.6), together with Eq. (2.6a), would result in summing contributions to the total *p-coefficients* from all data points. Instead, simplification is applied to result in a point-iteration approach, so that in each iteration over the dataset, the *p-coefficients* represent only the contribution from one data point. After simplification,

*r i i*

*r SSE*

1, ,

1 1, , ,

1 1, ,

( ), , ( )

*j k kj*

1 1 ,

*i i*

*product A x A x*

*product A x A x*

*i i*

*product A x A x x*

*product A x A x*

1 1, ,

*j k k j*

[ ( ), , ( )]

1 1 , ,

1 1, ,

*j k k j*

(2.7)

(2.8)

are the required precision criteria; and max *t* is a

[ ( ), , ( )]

*j j k j r k*

(2.6a)

(2.6)

(2.6b)

(2.6c)

(2.6d)

,1 ,

*y y p p*

*j j i i r r j*

*p p*

1

1

*i*

*i n*

0, 1 0, 0

,1 ,

satisfied (Wang, et al. 1998, 1999):

error) for simplicity in descriptions; 1

the first two criteria (Eqs. 2.7 and 2.8) are satisfied.

*p p yy*

*p p yy*

2) RER= <sup>1</sup>

where

Eq. (2.6) becomes

and for *r* = 1, 2, , *k*,

For *r* = 0,

where *i n* 1,2,..., the index of the cells, *n* is the total number of cells of the model; 1, 2, , , ( , ,, , ) *<sup>i</sup> Px x x x j j <sup>r</sup> <sup>j</sup> <sup>k</sup> <sup>j</sup>* is the internal function with parameters 0 1 , ,..., ,... *ii i i r k pp p p* to be determined, and , ( ) *<sup>i</sup> A x k k <sup>j</sup>* denotes the membership grade for *<sup>k</sup>*, *<sup>j</sup> x* . Each function covers a certain range of input variables.

#### **2.4 Defuzzification**

In each fuzzy cell, the contribution to the outcome (i.e. the cell output) is based on the internal function, Eq. (2.2). The final prediction of the outcome is the weighted average of all cell outputs after the process of reasoning algorithm. Because of this weighting among many factors over large ranges of possibilities, the word "fuzzy" is derived to describe the method. However, its prediction is never "fuzzy". The output is estimated by the center of gravity method. For the *j*th input ( 1, 2, , , , ,..., ,..., *j j <sup>r</sup> <sup>j</sup> <sup>k</sup> <sup>j</sup> xx x x* ), the output is as follows:

$$\hat{y}\_{j} = \frac{\sum\_{i=1}^{n} \text{product} \left[ A^{i}(\mathbf{x}\_{1,j}), \dots, A^{i}(\mathbf{x}\_{k,j}) \right] \mathbf{p}^{i}}{\sum\_{i=1}^{n} \text{product} \left[ A^{i}(\mathbf{x}\_{1,j}), \dots, A^{i}(\mathbf{x}\_{k,j}) \right]} \tag{2.3}$$

In Eq. (2.3) 1, , ( ),..., ( ) *i i <sup>j</sup> <sup>k</sup> <sup>j</sup> product A x A x* is the weighted factor of the *i*th cell; and the index *<sup>j</sup>* of the data set, where *j*=1,2,…, *m,* and *m* is the total number of the data records; and the "product" stands for product operator of its elements in this Chapter.

#### **2.5 Parameter identification**

Given a set of membership functions for each input variable, the unknown coefficients of the internal functions are determined by using the Newton gradient-descent method. The accuracy of the established aerodynamic model through the fuzzy-logic algorithm is estimated by the sum of squared errors (SSE) and the squared multiple correlation coefficients (*R2*):

$$SSE = \sum\_{j=1}^{m} (\hat{y}\_j - y\_j)^2 \tag{2.4}$$

$$R^2 = 1 - \frac{\{\sum\_{j=1}^m (\hat{y}\_j - y\_j)^2\}}{\{\sum\_{j=1}^m (\overline{y} - y\_j)^2\}}\tag{2.5}$$

In Eqs. (2.4) and (2.5), where ˆ*<sup>j</sup> y* , the output of the fuzzy-logic model at data point *j*, is estimated by Eq. (2.3); *yj* is the data point used for the model training at point *j*; *y* is the mean of the sample data, and m is the total number of data points. The model training is to determine the unknown coefficients of the internal functions, pr i , by maximizing the value of *R2*. These coefficients are determined by the following iterative formula to minimize the sum of squared error (Eq. 2.4):

$$p\_{r,t+1}^i = p\_{r,t}^i - \alpha\_r \frac{\partial (SSE)}{\partial p\_r^i} \tag{2.6}$$

$$\frac{\partial \langle SSE \rangle}{\partial p\_r^i} = 2 \sum\_{j=1}^m (\hat{y}\_j - y\_j) \frac{\partial \hat{y}\_j (\mathbf{x}\_{1,j}, \dots, \mathbf{x}\_{k,j}, p\_r^i, \dots, p\_k^n)}{\partial p\_r^i} \tag{2.6a}$$

$$\frac{\partial \hat{y}\_j}{\partial p\_r^i} = \frac{\text{product} [A\_1^i(\mathbf{x}\_{1,j}), \dots, A\_k^i(\mathbf{x}\_{k,j})] \mathbf{x}\_{r,j}}{\sum\_{i=1}^n \text{product} [A\_1^i(\mathbf{x}\_{1,j}), \dots, A\_k^i(\mathbf{x}\_{k,j})]} \tag{2.6b}$$

where*<sup>r</sup>* is the convergence factor or the step size in the gradient method; subscript index *t* denotes the iteration sequence, and *<sup>r</sup>*, *<sup>j</sup> x* =1.0 if r=0 in Eq. (2.6b). Usually, the magnitude of *<sup>r</sup>* is chosen based on that of the gradient. Eq. (2.6), together with Eq. (2.6a), would result in summing contributions to the total *p-coefficients* from all data points. Instead, simplification is applied to result in a point-iteration approach, so that in each iteration over the dataset, the *p-coefficients* represent only the contribution from one data point. After simplification, Eq. (2.6) becomes

For *r* = 0,

124 Fuzzy Logic – Emerging Technologies and Applications

where *i n* 1,2,..., the index of the cells, *n* is the total number of cells of the model;

In each fuzzy cell, the contribution to the outcome (i.e. the cell output) is based on the internal function, Eq. (2.2). The final prediction of the outcome is the weighted average of all cell outputs after the process of reasoning algorithm. Because of this weighting among many factors over large ranges of possibilities, the word "fuzzy" is derived to describe the method. However, its prediction is never "fuzzy". The output is estimated by the center of

*<sup>i</sup> A x k k <sup>j</sup>* denotes the membership grade for *<sup>k</sup>*, *<sup>j</sup> x* . Each function covers a

1, ,

*j k j*

( ), , ( )

*<sup>n</sup> i ii*

*product A x A x*

of the data set, where *j*=1,2,…, *m,* and *m* is the total number of the data records; and the

Given a set of membership functions for each input variable, the unknown coefficients of the internal functions are determined by using the Newton gradient-descent method. The accuracy of the established aerodynamic model through the fuzzy-logic algorithm is estimated by the sum of squared errors (SSE) and the squared multiple correlation

1

*m*

*j m*

1

In Eqs. (2.4) and (2.5), where ˆ*<sup>j</sup> y* , the output of the fuzzy-logic model at data point *j*, is estimated by Eq. (2.3); *yj* is the data point used for the model training at point *j*; *y* is the mean of the sample data, and m is the total number of data points. The model training is to

*R2*. These coefficients are determined by the following iterative formula to minimize the sum

*j*

*j SSE y y* 

2 1

1

*R*

determine the unknown coefficients of the internal functions, pr

( ) <sup>ˆ</sup> *<sup>m</sup> j j*

{ ( )} ˆ

*y y*

*j j*

{ ( )}

*y y*

*product A x A x p*

1, ,

2

2

2

*j*

(2.4)

i

, by maximizing the value of

*j k j*

( ), , ( )

*<sup>j</sup> <sup>k</sup> <sup>j</sup> product A x A x* is the weighted factor of the *i*th cell; and the index *<sup>j</sup>*

*r k pp p p* to be

(2.3)

(2.5)

*<sup>i</sup> Px x x x j j <sup>r</sup> <sup>j</sup> <sup>k</sup> <sup>j</sup>* is the internal function with parameters 0 1 , ,..., ,... *ii i i*

gravity method. For the *j*th input ( 1, 2, , , , ,..., ,..., *j j <sup>r</sup> <sup>j</sup> <sup>k</sup> <sup>j</sup> xx x x* ), the output is as follows:

*<sup>j</sup> <sup>n</sup> i i*

1

*i*

ˆ

*y*

*i i*

In Eq. (2.3) 1, , ( ),..., ( )

**2.5 Parameter identification** 

coefficients (*R2*):

of squared error (Eq. 2.4):

1

"product" stands for product operator of its elements in this Chapter.

*i*

1, 2, , , ( , ,, , )

certain range of input variables.

determined, and , ( )

**2.4 Defuzzification** 

$$p\_{0,t+1}^i = p\_{0,t}^i - 2a\_0(\hat{y}\_j - y\_j) \times \frac{\text{product}[A\_1^i(\mathbf{x}\_1), \dots, A\_k^i(\mathbf{x}\_{k,j})]}{\sum\_{s=1}^n \text{product}[A\_1^s(\mathbf{x}\_{1,j}), \dots, A\_k^s(\mathbf{x}\_{k,j})]} \tag{2.6c}$$

and for *r* = 1, 2, , *k*,

$$p\_{r,t+1}^i = p\_{r,t}^i - 2\alpha\_r(\hat{y}\_j - y\_j) \times \frac{\operatorname{product}[A\_1^i(\mathbf{x}\_1), \dots, A\_k^i(\mathbf{x}\_{k,j})] \mathbf{x}\_{r,j}}{\sum\_{s=1}^n \operatorname{product}[A\_1^s(\mathbf{x}\_{1,j}), \dots, A\_k^s(\mathbf{x}\_{k,j})]} \tag{2.6d}$$

The iteration during the search sequence stops when one of the following three criteria is satisfied (Wang, et al. 1998, 1999):

$$\begin{pmatrix} 1 \\ \end{pmatrix} \mathbf{Cost} = SSE\_t < \varepsilon\_1 \tag{2.7}$$

$$\text{2) }\text{RER} = \frac{SSE\_t - SSE\_{t-1}}{SSE\_t} < \varepsilon\_2 \tag{2.8}$$

$$\mathfrak{B} \nmid t = t\_{\text{max}} \tag{2.9}$$

In the above criteria, Cost= *SSEt* is the sum of squared errors (SSE) in current iteration to be denoted by "Cost" and RER=(cost\_current - cost\_previous)/cost\_current (i.e. the relative error) for simplicity in descriptions; 1 and <sup>2</sup> are the required precision criteria; and max *t* is a specified maximum iteration number. The convergence of modeling is achieved only when the first two criteria (Eqs. 2.7 and 2.8) are satisfied.

In application, the more complex the problem is, the more usefulness of the present algorithm would clearly exhibit. A complex problem in identifying the aerodynamic models of a jet transport in severe atmospheric turbulence will be presented in Section 4. Here, a simpler problem, yet complex enough for a conventional parameter identification method, will be used to show the robustness and reliability of the present algorithm. The idea is to assume the aerodynamic derivatives are known from the wind-tunnel forced oscillation test and the aerodynamic model data are generated from these derivatives. The algorithm is to obtain a numerical model containing the p-coefficients and the aerodynamic derivatives are

For this purpose, the rolling moment coefficient model will be examined. It is assumed to be

, , , p, r, k,

sin ( ) *<sup>n</sup>*

sin *<sup>n</sup>*

cos ( ) *<sup>n</sup>*

 *t*

 *t*

where k = b/2V, the reduced frequency and is the oscillation frequency in the test.

Define the rolling motion, (t), in wind-tunnel testing being described as follows.

 

 

*p*

(3.1)

Fig. 3. Identification process for the best structure

then estimated for comparison with the test data.

**3. Some verification examples** 

a function of

**3.1 Wind-tunnel data** 

Given membership functions and the training data, this parameter identification procedure can be applied to establish a fuzzy-logic model, i.e. determining the p-coefficients in Eq. 2.2. One important reason for the fuzzy logic algorithm, as described above, to work well in nonlinear, robust interpolation is that it employs numerous internal functions to cover the whole ranges of input variables.

#### **2.6 Model structure identification**

In the fuzzy-logic model, the model structure is indicated by the number of membership functions for each variable. For a fuzzy-logic model with multiple variables, the structure is the combination of the numbers and forms of the membership functions assigned to all input variables. Since the sequence defines the one-to-one relationship between the numbers and the forms for each variable, the structure can be uniquely described by numbers of the membership functions.

The model structure is determined by maximizing the correlation coefficient, Eq. (2.5). A search forward algorithm has been employed for the identification. At each search stage, there may be many fuzzy-logic models with different structure combinations. The search stage numbers are denoted by *Ns* . Out of all the possible intermediate fuzzy-logic models at each search stage, for an efficient search, only some structures are developed and evaluated. Two selection criteria, to be given below, are used to choose these structures. With the incremental sequence and the selection criteria, the search forward algorithm is summarized as follows (Wang, et al. 1998):


10 20 0 0 ( , , , , , 1) *NN N N r k* . Perform the identification of internal coefficients in Eq. (2.1) for each child structure and then calculate the *R2* by using Eq. (2.5);


The above process is illustrated in Fig. 3. In the structure identification, parameter identification to determine the p-parameters according to Eq. (2.6) is also needed. But the number of iteration to determine the p-parameters is limited to 2000, so that the best structure is decided on a relative basis. After this last step, Eq. (2.6) is applied iteratively until both the values of *R2* and RER reach the requirements in the final parameter identification.

Fig. 3. Identification process for the best structure

#### **3. Some verification examples**

126 Fuzzy Logic – Emerging Technologies and Applications

Given membership functions and the training data, this parameter identification procedure can be applied to establish a fuzzy-logic model, i.e. determining the p-coefficients in Eq. 2.2. One important reason for the fuzzy logic algorithm, as described above, to work well in nonlinear, robust interpolation is that it employs numerous internal functions to cover the

In the fuzzy-logic model, the model structure is indicated by the number of membership functions for each variable. For a fuzzy-logic model with multiple variables, the structure is the combination of the numbers and forms of the membership functions assigned to all input variables. Since the sequence defines the one-to-one relationship between the numbers and the forms for each variable, the structure can be uniquely described by numbers of the

The model structure is determined by maximizing the correlation coefficient, Eq. (2.5). A search forward algorithm has been employed for the identification. At each search stage, there may be many fuzzy-logic models with different structure combinations. The search stage numbers are denoted by *Ns* . Out of all the possible intermediate fuzzy-logic models at each search stage, for an efficient search, only some structures are developed and evaluated. Two selection criteria, to be given below, are used to choose these structures. With the incremental sequence and the selection criteria, the search forward algorithm is summarized

2. Assume an initial structure, also called parent structure as 10 20 0 0 ( , ,, ,, ) *NN N N r k* ; 3. Begin at the search stage number *Ns* =l , form all possible structures starting from the parent structure by adding one more membership function a time only to one input variable. Those all possible structures are called child structures as

10 20 0 0 ( , , , , , 1) *NN N N r k* . Perform the identification of internal coefficients in Eq.

4. Select the top 5 child structures among all calculated values of *R2*as new parent

5. Go back to step 2) starting from the new parent structures and repeat the same

6. Pick out the maximum value of *R2*among the child structures in each searching stage as <sup>2</sup> *R*max . The structure with the largest <sup>2</sup> *R*max corresponding to all picking values is the

The above process is illustrated in Fig. 3. In the structure identification, parameter identification to determine the p-parameters according to Eq. (2.6) is also needed. But the number of iteration to determine the p-parameters is limited to 2000, so that the best structure is decided on a relative basis. After this last step, Eq. (2.6) is applied iteratively until both the values of *R2* and RER reach the requirements in the final parameter

1. Specify the input variables *<sup>r</sup> x* , *r* = 1, 2,…, *k* and the output variable *y*;

10 20 0 0 ( 1, , , , , ) *NNNN r k* , 10 20 0 0 ( , 1, , , , ) *NN N N r k* , ,

procedures in steps 2) and 3) until the best structure is identified;

structures for next search step *Ns* = *Ns* + l ;

optimal structure within a sensible *Ns* .

identification.

(2.1) for each child structure and then calculate the *R2* by using Eq. (2.5);

whole ranges of input variables.

membership functions.

as follows (Wang, et al. 1998):

**2.6 Model structure identification** 

In application, the more complex the problem is, the more usefulness of the present algorithm would clearly exhibit. A complex problem in identifying the aerodynamic models of a jet transport in severe atmospheric turbulence will be presented in Section 4. Here, a simpler problem, yet complex enough for a conventional parameter identification method, will be used to show the robustness and reliability of the present algorithm. The idea is to assume the aerodynamic derivatives are known from the wind-tunnel forced oscillation test and the aerodynamic model data are generated from these derivatives. The algorithm is to obtain a numerical model containing the p-coefficients and the aerodynamic derivatives are then estimated for comparison with the test data.

For this purpose, the rolling moment coefficient model will be examined. It is assumed to be a function of

$$\mathbf{a}, \mathfrak{B}, \mathfrak{ϕ}, \mathfrak{p}, \mathfrak{r}, \mathbf{k}, \; \dot{\mathfrak{J}} \tag{3.1}$$

where k = b/2V, the reduced frequency and is the oscillation frequency in the test.

#### **3.1 Wind-tunnel data**

Define the rolling motion, (t), in wind-tunnel testing being described as follows.

$$\begin{aligned} \beta &= \alpha\_n \sin \phi(t) \\\\ \dot{\beta} &= p \sin \alpha\_n \\\\ \alpha &= \alpha\_n \cos \phi(t) \end{aligned}$$

Based on Eq. (3.4), two sets of data are generated at k=0.12 and 0.08, which are then combined into one for modeling. In linear aerodynamic theory, the rolling moment coefficient is known to be independent of the roll angle (). Two models are set up with = 0 or without in Eq. (3.1), and given by Eq. (3.2) to test the robustness of the algorithm. To calculate the response, input data in the form of Eq. (3.4) for a cosine harmonic oscillation are prepared. The output from the model is then Fourier-analyzed to obtain the in-phase and out-of-phase response. The out-of-phase response is the damping component and is what to be presented below. Only the small-amplitude results are presented, because the

= -0.0688; *(Clp)osc* = -0.1736; *(Clr)osc* = 0.0999

 = -0.0688; *(Clp)osc* = -0.1736; *(Clr)osc* = 0.0999 It is seen that the results are identical at = 5 degrees in both cases, and agree with the original wind-tunnel data very well. Same results are obtained if is absent in the model

= -0.2493; *(Clp)osc* = -0.1581; *(Clr)osc* = 0.3926

 = -0.2494; *(Clp)osc* = -0.1584; *(Clr)osc* = 0.3925 The results at = 20 degrees are nearly identical, except the last digit and also agree with

The large-amplitude test cases produce similarly accurate results as compared with the

**3.1.2 Large amplitudes** 

**3.2 Modeling results** 

structure.

the original data well.

**3.2.3 Large-amplitude test cases** 

**3.1.2.1 = 5 deg., = 30 deg., = 15 deg., k=0.12 3.1.2.2 = 20 deg.; =15 deg., = 15 deg., k=0.08** 

large-amplitude results are very similar.

**3.2.1 = 5 deg., = 5 deg., = 5 deg., k=0.12** 

*Cl*

*Cl*

**3.2.2 = 20 deg.; =5 deg., = 5 deg., k=0.08** 

*Cl*

*Cl*

If = 0 or it is absent in the model data, the modeling results are:

wind-tunnel data. Therefore, the results will not be repeated.

Assume (t) 0 and is given by Eq. (3.2). The modeling results are:

On the other hand, if is assumed 0 in the model data, the modeling results are:

Again, assume (t) 0 and is calculated with Eq. (3.2). The modeling results are:

All derivatives are taken to be the same as in the small amplitude case.

$$\phi(t) = -\Delta\phi\cos(k\overline{t})\tag{3.2}$$

where n is the nominal angle of attack used in wind tunnel testing and p is the roll rate. Since the rolling moment coefficient is also affected by yawing motion, ψ(t), the latter in wind-tunnel testing is assuming to be:

$$
\boldsymbol{\beta} = -\psi \cos \alpha\_n
$$

$$
\dot{\boldsymbol{\beta}} = -\psi \cos \alpha\_n \,\,\,\,
$$

$$
\psi = r \,\,\,\,\,\text{the yaw rate}
$$

$$
\boldsymbol{\alpha} = \alpha\_n
$$

$$
\psi(t) = -\Delta \,\,\psi \cos(k\overline{t})\tag{3.3}
$$

According to the linear theory, the rolling moment coefficient is calculated from:

$$\mathbf{C}\_{l} = \mathbf{C}\_{l\beta}\boldsymbol{\beta} + \mathbf{C}\_{l\eta}\overline{\boldsymbol{p}} + \mathbf{C}\_{lr}\overline{\boldsymbol{r}} + \mathbf{C}\_{l\dot{\beta}}\overline{\boldsymbol{\beta}}\tag{3.4}$$

where the bar over a variable indicate a dimensionless one. For example, *p pb V* / 2 , where V is the airspeed. The verification will be performed at two conditions: one at small oscillation amplitudes, and the other one at large amplitude. These conditions are given in the following.

 

#### **3.1.1 Small amplitudes at two reduced frequencies**

#### **3.1.1.1 = 5 deg., = 5 deg., = 5 deg., k=0.12**

The wind-tunnel data for a test model show: *Cl* = -0.0688; *Clp*= -0.17; *Clr* = 0.06; *<sup>l</sup> C* =-0.04

Therefore,

$$\begin{aligned} (\mathbf{C}\_{lp})\_{\mathrm{esc}} &= \mathbf{C}\_{lp} + \mathbf{C}\_{l\dot{\beta}} \sin \alpha = \text{-0.1735} \\\\ (\mathbf{C}\_{lr})\_{\mathrm{esc}} &= \mathbf{C} \text{l} \mathbf{r} - \mathbf{C}\_{l\dot{\beta}} \cos \alpha = \text{0.0998} \end{aligned}$$

#### **3.1.1.2 = 20 deg.; =5 deg., = 5 deg., k=0.08**

The wind-tunnel data show: *Cl* = -0.2493; *Clp*= -0.10; *Clr* = 0.233; *<sup>l</sup> C* =-0.17

Therefore,

$$\begin{aligned} \text{(C}\_{l\text{p}}\text{)}\_{\text{esc}} &= \text{C}\_{l\text{p}} + \text{C}\_{l\dot{\beta}} \sin\alpha = \text{-0.1581} \\\\ \text{(C}\_{l\text{r}}\text{)}\_{\text{esc}} &= \text{C}\_{l\text{r}} - \text{C}\_{l\dot{\beta}} \cos\alpha = 0.3927 \end{aligned}$$

#### **3.1.2 Large amplitudes**

128 Fuzzy Logic – Emerging Technologies and Applications

 

where n is the nominal angle of attack used in wind tunnel testing and p is the roll rate. Since the rolling moment coefficient is also affected by yawing motion, ψ(t), the latter in

cos

cos

*r* , the yaw rate

 

*l l lp lr <sup>l</sup> C C Cp Cr C*

where the bar over a variable indicate a dimensionless one. For example, *p pb V* / 2 , where V is the airspeed. The verification will be performed at two conditions: one at small oscillation amplitudes, and the other one at large amplitude. These conditions are given in

= -0.2493; *Clp*= -0.10; *Clr* = 0.233; *<sup>l</sup> C*

sin = -0.1735

cos = 0.0998

sin = -0.1581

cos = 0.3927

 *n*

( ) cos( ) *t k t* (3.2)

( ) cos( ) *t k t* (3.3)

= -0.0688; *Clp*= -0.17; *Clr* = 0.06; *<sup>l</sup> C*

=-0.17

(3.4)

=-0.04

 *n*

 *<sup>n</sup>* ,

According to the linear theory, the rolling moment coefficient is calculated from:

*(Clp)osc*= *Clp* + *<sup>l</sup> C*

*(Clr)osc* = Clr - *<sup>l</sup> C*

*(Clp)osc*= *Clp* + *<sup>l</sup> C*

*(Clr)osc* = *Clr* - *<sup>l</sup> C*

**3.1.1 Small amplitudes at two reduced frequencies** 

**3.1.1.1 = 5 deg., = 5 deg., = 5 deg., k=0.12** 

**3.1.1.2 = 20 deg.; =5 deg., = 5 deg., k=0.08** 

The wind-tunnel data show: *Cl*

The wind-tunnel data for a test model show: *Cl*

wind-tunnel testing is assuming to be:

the following.

Therefore,

Therefore,

#### **3.1.2.1 = 5 deg., = 30 deg., = 15 deg., k=0.12**

#### **3.1.2.2 = 20 deg.; =15 deg., = 15 deg., k=0.08**

All derivatives are taken to be the same as in the small amplitude case.

#### **3.2 Modeling results**

Based on Eq. (3.4), two sets of data are generated at k=0.12 and 0.08, which are then combined into one for modeling. In linear aerodynamic theory, the rolling moment coefficient is known to be independent of the roll angle (). Two models are set up with = 0 or without in Eq. (3.1), and given by Eq. (3.2) to test the robustness of the algorithm. To calculate the response, input data in the form of Eq. (3.4) for a cosine harmonic oscillation are prepared. The output from the model is then Fourier-analyzed to obtain the in-phase and out-of-phase response. The out-of-phase response is the damping component and is what to be presented below. Only the small-amplitude results are presented, because the large-amplitude results are very similar.

#### **3.2.1 = 5 deg., = 5 deg., = 5 deg., k=0.12**

Assume (t) 0 and is given by Eq. (3.2). The modeling results are:

$$C\_{l\beta} = \text{-0.0688}; \ (C\_{l\eta})\_{\text{osc}} = \text{-0.1736}; \ (C\_{lr})\_{\text{osc}} = \text{0.0999}$$

On the other hand, if is assumed 0 in the model data, the modeling results are:

$$C\_{l\beta} = -0.06888; \ (C\_{lp})\_{osc} = -0.1736; \ (C\_{lr})\_{osc} = 0.0999$$

It is seen that the results are identical at = 5 degrees in both cases, and agree with the original wind-tunnel data very well. Same results are obtained if is absent in the model structure.

#### **3.2.2 = 20 deg.; =5 deg., = 5 deg., k=0.08**

Again, assume (t) 0 and is calculated with Eq. (3.2). The modeling results are:

$$C\_{l\beta} = -0.249\Re \text{; } (C\_{l\eta})\_{osc} = -0.1581; \ (C\_{lr})\_{osc} = 0.3926$$

If = 0 or it is absent in the model data, the modeling results are:

$$C\_{l\beta} = \text{-0.2494}; \ (C\_{l\eta})\_{osc} = \text{-0.1584}; \ (C\_{lr})\_{osc} = \text{0.3925}$$

The results at = 20 degrees are nearly identical, except the last digit and also agree with the original data well.

#### **3.2.3 Large-amplitude test cases**

The large-amplitude test cases produce similarly accurate results as compared with the wind-tunnel data. Therefore, the results will not be repeated.

The twin-jet transport in the present study encountered clear-air turbulence in cruise flight at the altitude around 10,050 m. As a result, several passengers and cabin crews sustained injuries, because of which this event was classified as an accident. The present study was initiated to examine possible concepts of accident prevention in the future. The dataset used for the modeling are extracted from the FDR during turbulence encounter lasting for 92

The main aircraft geometric and inertial characteristics are taken, or estimated, as shown in

*W* (take-off) 1,431,800 N (321900 lb) *Ixx* 10,710,000 kg-m2 (7,899,900 slugs-ft2) *S* 260 m2 (2798.7 ft2) *Iyy* 14,883,800 kg-m2 (10,978,000 slugs-ft2) *c* 6.608 m (21.68 ft) *Izz* 25,283,271 kg-m2 (18,648,470 slugs-ft2)

The required operational parameters in FDR dataset for generating aerodynamic model data

), longitudinal acceleration (*ax*), lateral acceleration (*ay*), vertical acceleration (*az*), angle of

air temperature, wind speed, wind direction, and fuel flow rate. Since only the normal acceleration is recorded in 8-Hz resolution (i.e. 8 points per second), all other parameters are interpolated with a monotone cubic spline to the same sampling rate. Based on the principle in flight data analysis, to estimate stability (or sensitivity) derivative with a flight variable, the corresponding flight variable must be sufficiently excited in the flight. This principle can be satisfied by choosing a large time period so that flight variables have sufficient variation during the time period, or by combining different flights if a model to represent a particular

Typically, the longitudinal, lateral, and vertical accelerations (*ax*, *ay*, *az*) along the (x, y, z)-

control deflections are available and recorded in the FDR of all transport aircraft. Since the recorded flight data may contain errors (or called biases), compatibility analysis is

, and the Euler angles (

 (4.1)

> 

 

 *pq r* 

> *q r*

 ( sin cos )sec *q r* 

performed to remove them by satisfying the following kinematic equations:

 sin tan cos tan 

cos sin

*e*), rudder (

), pitch attitude (

*r*), stabilizer (

, , and 

(4.2)

(4.3)

), magnetic heading

), as well as all

*<sup>s</sup>*), engine EPR, outside

*b* 44.827 m (147.08 ft) *Ixz* 0.0 kg-m2 (0.0 slugs-ft2)

*a*), elevator (

**4. Application to aircraft aerodynamic modeling** 

Geometric data Moments of inertia

Table 1. The main aircraft geometric and inertial characteristics

are time (*t*), CAS, pressure altitude (*h*), roll attitude (

), aileron deflection (

**4.1 Flight data** 

seconds.

Table 1:

(

attack (

aircraft is desired.

**4.2 Compatibility analysis** 

body axes of aircraft, angle of attack

#### **3.2.4 Concluding remarks**

In the above example the model prediction practically shows the same results as the windtunnel data with or without the extra ϕ(t)-variable in the model. It illustrates one important concept in the present fuzzy logic aerodynamic modeling that more variables than what are known in the present linear theory may be included in the model without affecting the results of prediction. In the case of nonlinear theory, including more variables in the model allows presently unknown phenomena to be captured at the expense of more computing time.

#### **3.3 Modeling of wind-tunnel unsteady aerodynamic data**

Verification with other methods is difficult to conduct because of the unavailability of suitable data and published results. However, the present algorithm has been verified with wind-tunnel experimental data. The wind-tunnel data used consist of static, forced oscillation, and some cases with rotary balance data, in numerous sets. These data sets at various angles of attack and reduced frequencies are combined to set up six (6) aerodynamic models. The resulting models can predict aerodynamic hysteresis quite well (Wang, et al. 1998, 1999). To save space, all these correlation results will not be presented, except one pitching moment curve. Fig. 4 presents the comparison of experimental data and modeling prediction. As indicated earlier, the modeling results predict only the mean approximation in the least-square sense and are seen here to re-produce well the hysteresis in the test data. Note that k is defined as *c* /V in this case. The arrows indicate the direction of changes in Cm as varies in Fig. 4. As will be explained later, if the hysteresis curve is counterclockwise, as shown at low 's, the oscillatory pitch damping derivative is stable (i.e. negative in sign). On the other hand, if it is clockwise, as shown at high 's, the damping derivative is unstable (i.e. positive in sign).

Fig. 4. Comparison of experimental forced oscillation data with modeling results in pitching moment coefficient

### **4. Application to aircraft aerodynamic modeling**

#### **4.1 Flight data**

130 Fuzzy Logic – Emerging Technologies and Applications

In the above example the model prediction practically shows the same results as the windtunnel data with or without the extra ϕ(t)-variable in the model. It illustrates one important concept in the present fuzzy logic aerodynamic modeling that more variables than what are known in the present linear theory may be included in the model without affecting the results of prediction. In the case of nonlinear theory, including more variables in the model allows presently unknown phenomena to be captured at the expense of more computing

Verification with other methods is difficult to conduct because of the unavailability of suitable data and published results. However, the present algorithm has been verified with wind-tunnel experimental data. The wind-tunnel data used consist of static, forced oscillation, and some cases with rotary balance data, in numerous sets. These data sets at various angles of attack and reduced frequencies are combined to set up six (6) aerodynamic models. The resulting models can predict aerodynamic hysteresis quite well (Wang, et al. 1998, 1999). To save space, all these correlation results will not be presented, except one pitching moment curve. Fig. 4 presents the comparison of experimental data and modeling prediction. As indicated earlier, the modeling results predict only the mean approximation in the least-square sense and are seen here to re-produce well the hysteresis in the test data. Note that k is defined as *c* /V in this case. The arrows indicate the direction of changes in Cm as varies in Fig. 4. As will be explained later, if the hysteresis curve is counterclockwise, as shown at low 's, the oscillatory pitch damping derivative is stable (i.e. negative in sign). On the other hand, if it is clockwise, as shown at high 's, the damping

Fig. 4. Comparison of experimental forced oscillation data with modeling results in pitching

**3.3 Modeling of wind-tunnel unsteady aerodynamic data** 

derivative is unstable (i.e. positive in sign).

moment coefficient

**3.2.4 Concluding remarks** 

time.

The twin-jet transport in the present study encountered clear-air turbulence in cruise flight at the altitude around 10,050 m. As a result, several passengers and cabin crews sustained injuries, because of which this event was classified as an accident. The present study was initiated to examine possible concepts of accident prevention in the future. The dataset used for the modeling are extracted from the FDR during turbulence encounter lasting for 92 seconds.

The main aircraft geometric and inertial characteristics are taken, or estimated, as shown in Table 1:


Table 1. The main aircraft geometric and inertial characteristics

The required operational parameters in FDR dataset for generating aerodynamic model data are time (*t*), CAS, pressure altitude (*h*), roll attitude (), pitch attitude (), magnetic heading (), longitudinal acceleration (*ax*), lateral acceleration (*ay*), vertical acceleration (*az*), angle of attack (), aileron deflection (*a*), elevator (*e*), rudder (*r*), stabilizer (*<sup>s</sup>*), engine EPR, outside air temperature, wind speed, wind direction, and fuel flow rate. Since only the normal acceleration is recorded in 8-Hz resolution (i.e. 8 points per second), all other parameters are interpolated with a monotone cubic spline to the same sampling rate. Based on the principle in flight data analysis, to estimate stability (or sensitivity) derivative with a flight variable, the corresponding flight variable must be sufficiently excited in the flight. This principle can be satisfied by choosing a large time period so that flight variables have sufficient variation during the time period, or by combining different flights if a model to represent a particular aircraft is desired.

#### **4.2 Compatibility analysis**

Typically, the longitudinal, lateral, and vertical accelerations (*ax*, *ay*, *az*) along the (x, y, z) body axes of aircraft, angle of attack , and the Euler angles (, , and ), as well as all control deflections are available and recorded in the FDR of all transport aircraft. Since the recorded flight data may contain errors (or called biases), compatibility analysis is performed to remove them by satisfying the following kinematic equations:

$$\dot{\phi} = p + q \sin \phi \tan \theta + r \cos \phi \tan \theta \tag{4.1}$$

$$
\dot{\theta} = q \cos \phi - r \sin \phi \tag{4.2}
$$

$$
\psi = (q\sin\phi + r\cos\phi)\sec\theta \tag{4.3}
$$

C*<sup>m</sup> q S c* = *Iyy q* – *Ixz* (*r2* – *p2* ) – (*Izz* –*Ixx*)*rp* – *Tm* (4.16)

C*<sup>n</sup> q S b* = *Izz r* – *Ixz*( *p* –*qr*) – (*Ixx*–*Iyy*)*pq* (4.17)

where *m* is the aircraft mass; *q* the dynamic pressure; *S* the wing reference area; *Cx*, *Cz*, and *Cm* the longitudinal aerodynamic force and moment coefficients; *Cy*, *Cl*, and C*n* the lateraldirectional aerodynamic force and moment coefficients; *Ixx*, *Iyy*, and *Izz* the moments of inertia about *x*-, *y*-, and *z*-axes, respectively; *Ixy*, *Ixz*, and *Iyz* the products of inertia; and *Tx*, *Ty*, *Tz*, and *Tm* the thrust terms about *x*-, *y*-, *z*-axes, and in equation of pitching moment,

The above equations are used to determine all aerodynamic coefficients based on

and thrusts (*Tx*, *Ty*, *Tz*, and *Tm*). The angular rates are estimated through compatibility analysis. Since thrust was not measured during flight for most flight vehicles, those values and the effects on the forces and pitching moments in equations of (4.12), (4.13), (4.14), and

The reduced frequency is a parameter to indicate the degree of unsteadiness in unsteady aerodynamics and is estimated in this paper by fitting the local trajectory with a harmonic motion. In the static case, the reduced frequency is 0. Large values of the reduced frequency imply the importance of unsteady aerodynamic effect. For longitudinal aerodynamics, the equivalent harmonic motion is the one based on the angle-of-attack variation following the classical unsteady aerodynamic theory of Theodorsen (Theodorsen 1935). For lateraldirectional aerodynamics, it is based on the time variation of roll angle (Wang, et al. 1998).

, , and 

), angular rates (*p*, *q* and *r*),

) and time rate of angle

cos( ) *t* (4.18)

). These unknowns are calculated through an

), the local amplitude of the harmonic motion (*a*), the

 *a t*

2 2

*<sup>i</sup>* is the measured value at point *i* and *n* is the number of the data points

(4.20)

(4.19)

) is fitted with one of a harmonic motion at any instant as follows

 

sin( ) 

( cos( )) ( sin( ))

( )*t* = *a t* 

where those terms on the left hand side of Eqs. (4.18) and (4.19) are given and the unknowns

*i ii i*

used in the optimization. For the case in the present study, *n* =20 is found to be the best

respectively in Eqs. (4.12) ~ (4.17).

**4.3 Equivalent harmonic motion** 

of attack (d

phase lag (

/dt, or

In Eq. (4.20), where

are the local mean angle of attack (

1

*i*

*n*

(Wang, et al. 1998):

accelerometer readings (*ax*, *ay*, and *az*), Euler angles (

(4.16) should be predicted by a thrust model (see Section 4.4).

For the longitudinal motion, the time history of the angle of attack (

( )*t* = 

 

optimization method by minimizing the following cost function (least squares)

), and the angular frequency (

*J at* 

 ( sin )cos cos ( sin cos )sin *Vag x y a g* ( cos cos )sin cos *<sup>z</sup> a g* (4.4)

$$\dot{a} = \left[ (a\_z + \mathcal{g}\cos\theta\cos\phi)\cos a - (a\_x - \mathcal{g}\sin\theta)\sin a \right] / (V\cos\beta) \ + q - \tan\beta (p\cos a + r\sin a) \tag{4.5}$$

$$\dot{\beta} = \cos\beta (a\_y + g\cos\theta\sin\phi) / V + p\sin a - r\cos a$$

$$-\sin\beta [(a\_z + g\cos\theta\cos\phi)\sin a - (a\_x - g\sin\theta)\cos a] / V \tag{4.6}$$

where g is acceleration due to gravity, *V* is flight speed, is sideslip angle, *p* is roll rate, *q* is pitch rate, and *r* is yaw rate in Eqs. (4.1) ~ (4.6). Let the biases be denoted by , , ,,,, , , ,, , *xyz a a a pqrV b b b bbbb b b bbb* , respectively for *ax*, *ay*, *az*, etc. These biases are estimated by minimizing the squared sum of the differences between the two sides of the above equations. These equations in vector form can be written as:

$$
\dot{\vec{z}} = \vec{f}(\mathbf{x}) = \vec{f}(\mathbf{x}\_m - \Delta \mathbf{x}) \tag{4.7}
$$

where

$$\bar{z} = \begin{pmatrix} V, \alpha, \beta, \theta, \phi, \psi \end{pmatrix}^T \tag{4.8}$$

$$\bar{\mathbf{x}}\_m = \begin{pmatrix} a\_x, a\_y, a\_z, p, q, r, V, \alpha, \beta, \theta, \phi, \phi, \psi \end{pmatrix}^T \tag{4.9}$$

$$
\Delta \overline{\mathbf{x}} = \left( b\_{a\_x} b\_{a\_y}, b\_{a\_z} b\_{a\_z} b\_p, b\_{q'} b\_{r'} b\_{V'} b\_{a'} b\_{\beta'} b\_{\beta'} b\_{\phi'} b\_{\phi'} b\_{\phi'} \right)^T \tag{4.10}
$$

where the subscript "*m*" indicates the measured or recorded values. The cost function is defined as:

$$J = \frac{1}{2} (\dot{\bar{z}} - \bar{f})^T Q (\dot{\bar{z}} - \bar{f}) \tag{4.11}$$

where *Q* is a weighting diagonal matrix with elements being 1.0 except the one for the slowly varying flight speed being 10.0 and *z* is calculated with a central difference scheme with *zm* , which is the measured value of *<sup>z</sup>* . The steepest descent optimization method is adopted to minimize the cost function. As a result of the analysis, variables not present in the FDR, such as , *p*, *q* and *r*, are also estimated.

The force and moment coefficients are obtained from the following flight dynamic equations (Roskam 2003) about the airplane body axes:

$$
tau\_x = \mathbf{C}\_x \overline{\eta} \mathbf{S} + T\_x \tag{4.12}$$

$$
tau\_y = \mathbb{C}\_y \overline{q} \mathbf{S} + T\_y \tag{4.13}$$

$$
tau\_z = \mathbf{C}\_z \overline{\mathbf{q}} \mathbf{S} + T\_z \tag{4.14}$$

$$\mathbf{C} \cdot \overline{q} \text{ S } b = I\_{xx} \ \dot{p} - I\_{xz} (\dot{r} + pq) - (I\_{yy} - I\_{zz})qr \tag{4.15}$$

$$\mathbf{C}\_{m}\overline{q} \text{ S } \overline{c}\tag{4.15} \\ \mathbf{S} \,\overline{c}\,\overline{c}\,\, = I\_{yy} \,\, \dot{q} - I\_{xz} \left(r^2 - p^2\right) - \left(I\_{zz} - I\_{xx}\right)rp - T\_m \tag{4.16}$$

$$\mathbf{C}\_{n}\,\,\overline{q}\,\,\mathbf{S}\,\,\mathbf{b} = I\_{zz}\,\,\,\dot{\mathbf{r}} - I\_{xz}(\,\dot{\mathbf{p}}\,\,-q\mathbf{r}) - (I\_{xx} - I\_{yy})pq\,\tag{4.17}$$

where *m* is the aircraft mass; *q* the dynamic pressure; *S* the wing reference area; *Cx*, *Cz*, and

*Cm* the longitudinal aerodynamic force and moment coefficients; *Cy*, *Cl*, and C*n* the lateraldirectional aerodynamic force and moment coefficients; *Ixx*, *Iyy*, and *Izz* the moments of inertia about *x*-, *y*-, and *z*-axes, respectively; *Ixy*, *Ixz*, and *Iyz* the products of inertia; and *Tx*, *Ty*, *Tz*, and *Tm* the thrust terms about *x*-, *y*-, *z*-axes, and in equation of pitching moment, respectively in Eqs. (4.12) ~ (4.17).

The above equations are used to determine all aerodynamic coefficients based on accelerometer readings (*ax*, *ay*, and *az*), Euler angles (, , and ), angular rates (*p*, *q* and *r*), and thrusts (*Tx*, *Ty*, *Tz*, and *Tm*). The angular rates are estimated through compatibility analysis. Since thrust was not measured during flight for most flight vehicles, those values and the effects on the forces and pitching moments in equations of (4.12), (4.13), (4.14), and (4.16) should be predicted by a thrust model (see Section 4.4).

#### **4.3 Equivalent harmonic motion**

132 Fuzzy Logic – Emerging Technologies and Applications

 

 *a g ag V* (4.6)

 

*a g Vp r*

 

 *ag V q pr* tan ( cos sin )

> >

, respectively for *ax*, *ay*, *az*, etc. These biases are

(4.7)

(4.8)

*<sup>T</sup> J z <sup>f</sup> Q z <sup>f</sup>* (4.11)

(4.10)

*T*

is calculated with a central difference scheme

. The steepest descent optimization method is

*ma C qS T xx x* (4.12)

*ma C qS T yy y* (4.13)

*ma C qS T zz z* (4.14)

 

 

is sideslip angle, *p* is roll rate, *q* is

(4.4)

 (4.5)

 

 

 

*m xyz x a a a pqrV*

2

, *p*, *q* and *r*, are also estimated.

*a g* ( cos cos )sin cos *<sup>z</sup> a g*

cos ( cos sin ) / sin cos *<sup>y</sup>*

pitch rate, and *r* is yaw rate in Eqs. (4.1) ~ (4.6). Let the biases be denoted

estimated by minimizing the squared sum of the differences between the two sides of the

( , , ,,, )*<sup>T</sup> z V* 

( , , ,,,, , , , ,, )*<sup>T</sup>*

( , , ,,,, , , ,,,) *xyz*

<sup>1</sup> ( )( )

where *Q* is a weighting diagonal matrix with elements being 1.0 except the one for the

adopted to minimize the cost function. As a result of the analysis, variables not present in

The force and moment coefficients are obtained from the following flight dynamic equations

C*<sup>l</sup> q S b* = *Ixx p* – *Ixz*(*r* + *pq*) – *(Iyy* – *Izz*)*qr* (4.15)

where the subscript "*m*" indicates the measured or recorded values. The cost function is

*a a a pqrV x b b b bbbb b b bbb*

 (4.9)

( sin )cos cos ( sin cos )sin *Vag x y*

 

sin [( cos cos )sin ( sin )cos ]/ *z x*

where g is acceleration due to gravity, *V* is flight speed,

 

() ( ) *z f x f x x <sup>m</sup>*

 

[( cos cos )cos ( sin )sin ]/( cos ) *z x*

above equations. These equations in vector form can be written as:

 

 

by , , ,,,, , , ,, , *xyz a a a pqrV b b b bbbb b b bbb*

slowly varying flight speed being 10.0 and *z*

(Roskam 2003) about the airplane body axes:

, which is the measured value of *<sup>z</sup>*

where

defined as:

with *zm*

the FDR, such as

*a g*

The reduced frequency is a parameter to indicate the degree of unsteadiness in unsteady aerodynamics and is estimated in this paper by fitting the local trajectory with a harmonic motion. In the static case, the reduced frequency is 0. Large values of the reduced frequency imply the importance of unsteady aerodynamic effect. For longitudinal aerodynamics, the equivalent harmonic motion is the one based on the angle-of-attack variation following the classical unsteady aerodynamic theory of Theodorsen (Theodorsen 1935). For lateraldirectional aerodynamics, it is based on the time variation of roll angle (Wang, et al. 1998).

For the longitudinal motion, the time history of the angle of attack () and time rate of angle of attack (d/dt, or ) is fitted with one of a harmonic motion at any instant as follows (Wang, et al. 1998):

$$a(\mathbf{t}) \equiv \overline{a} + \alpha \cos(\alpha \mathbf{t} + \overline{\phi}) \tag{4.18}$$

$$
\dot{\alpha}(t) = -a\rho \sin(\alpha t + \overline{\phi}) \tag{4.19}
$$

where those terms on the left hand side of Eqs. (4.18) and (4.19) are given and the unknowns are the local mean angle of attack ( ), the local amplitude of the harmonic motion (*a*), the phase lag ( ), and the angular frequency ( ). These unknowns are calculated through an optimization method by minimizing the following cost function (least squares)

$$J = \sum\_{i=1}^{n} \left[ \alpha\_i - \left( \overline{\alpha} + a \cos(\alpha t\_i + \overline{\phi}) \right) \right]^2 + \left[ \dot{\alpha}\_i - \left( \overline{\alpha} + a \cos \text{in} (\alpha t\_i + \overline{\phi}) \right) \right]^2 \tag{4.20}$$

In Eq. (4.20), where*<sup>i</sup>* is the measured value at point *i* and *n* is the number of the data points used in the optimization. For the case in the present study, *n* =20 is found to be the best

 *T* = *f* (*h*, *W*, *M*, CAS, EPR, *mf* ) (4.23)

For GE or CFM turbofan engines, the rpm of the low-pressure compressor (*N1*) is used to set

 *T* = *f* (*h*, *W*, *M*, CAS, *N1*, *mf* ) (4.24)

In the present study, the P&W turbofan engines powering the twin-jet transport under study will be illustrated. The actual thrust in operation is obtained by using the recorded

The following climb equation (Lan & Roskam 2008) is to be satisfied in the least square sense

sin *W dV TDW g dt*

> *D D W L*

cos

All these equations are still valid in descent with negative climb angles (). The above equations are further employed for parameter identification in the process of modeling.

Once the thrust model is generated as a function of *h*, *W*, *M*, CAS, EPR, and *mf* with the flight conditions of climbing, cruise, and descent, one can estimate the thrust magnitude by

Modeling means to establish the numerical relationship among certain variables of interest. In the fuzzy-logic model, more complete necessary influencing flight variables can be included to capture all possible effects on aircraft response to atmospheric disturbances. For

where the left hand side represents the coefficients of axial force (*Cx*), normal force (*Cz*), and pitching moment (*Cm*), respectively. All variables on the right hand side of Eq. (4.27) have been defined in the previous section. It should be noted that the stabilizer angle (s) is included here, because it varies, though slowly, in flight to provide pitch trim (i.e. reducing the total static pitching moment to 0.0). The roll rate is included here because it is known that an aircraft under high aerodynamic loads at transonic speeds may have its longitudinal stability derivatives affected when additional disturbance due to roll rate is

, *e*, *M*, *p*, 

(4.25)

(4.26)

*<sup>s</sup>*, *q* ) (4.27)

the level of thrust, so that the thrust model is set up as:

variables in the FDR, in particular the fuel flow rates.

inserting these flight variables from the FDR into the model.

longitudinal aerodynamics, the models are assumed to be of the form:

,, *q*, *k1*,

**4.5 Fuzzy-Logic unsteady aerodynamic models** 

 *Cx, Cz, Cm* = *f* (

For the lateral-directional aerodynamics,

over a 5-second internal:

and

imposed.

choice by correlating with a cosine wave with a constant frequency. The 20 points preceding and including the current time are employed in Eq. (4.20). The least-square method is found to converge well and gives reasonably accurate results. The lateral-directional equivalent reduced frequency is computed in the same manner.

The local equivalent reduced frequency in the longitudinal motion is defined as,

$$k\_I = \frac{\alpha \overline{c}}{V} \tag{4.21}$$

where *c* is the mean chord length of wing airfoil section. The lateral-directional equivalent reduced frequency is defined as

$$k\_2 = \frac{\alpha b}{2V} \tag{4.22}$$

where *b* is the wing span.

#### **4.4 Fuzzy-Logic thrust model**

As shown before, the thrust terms appear in the force equations and the pitching moment equation (Eqs. 4.12~4.14 and 4.16; but in the current application, Ty =Tz = 0.). Since the values of thrust for aircraft in flight cannot be directly measured in the current state of the art, they are not recorded in the FDR. The manufacturers of engines agreed that using such parameters as the Mach number, airspeed, flight altitude, temperature, the rpm of the pressure compressors and engine pressure ratios is adequate to estimate the engine thrust. A realistic thrust model is quite complex and cannot be represented by any simple equation. Since such thrust model is not available for the present study, a realistic one tied to the recorded engine performance parameters is developed with the fuzzy-logic algorithm.

For a commercial aircraft, most likely only the axial force and the pitching moment are affected by thrust. This assumption will be made in this Chapter. Theoretically, clear-air turbulence (i.e. random change in u, w (or ) and v (or )) affect the engine performance through its effects on static and dynamic distortions at the engine face. However, its effects are not known and cannot be estimated, and therefore ignored in the present application.

For the present purpose, data from the flight manual for the fuel flow rates ( *mf* ) at various

altitudes (*h*), weights (*W*), Mach numbers (*M*), calibrated airspeed (CAS), engine pressure ratios (EPR), in cruise flight are utilized. Note that the drag polar for a given aircraft is generally not known to most researchers. To estimate it and hence the thrust magnitude in cruise, the assumption of a design lift-to-drag ratio (*L/D*) of 17.5 is made. This value of liftto-drag in cruise is assumed based on the past design experience for twin-jet transports. In the flight manual, various weights, altitudes, Mach numbers, CAS, EPR, and fuel flow rates in cruise are tabulated. The lift coefficient can be calculated at each flight condition immediately. As a result, the drag coefficient can be estimated from the assumption of liftto-drag ratio. Therefore, the design thrust in cruise at various Mach numbers can be estimated. For the Pratt & Whitney turbofan engines, thrust (*T*) is defined by EPR, so that the thrust model is set up as:

$$T = f\left(\text{h, } \mathcal{W}\_{\prime} \mathcal{M}\_{\prime} \text{CAS}\_{\prime} \text{ EPR}, \dot{m}\_{f}\right) \tag{4.23}$$

For GE or CFM turbofan engines, the rpm of the low-pressure compressor (*N1*) is used to set the level of thrust, so that the thrust model is set up as:

$$T = f\left(\text{lb, } \mathcal{W}, \mathcal{M}, \text{CAS, } \mathcal{N}\_1, \dot{\mathcal{m}}\_f\right) \tag{4.24}$$

In the present study, the P&W turbofan engines powering the twin-jet transport under study will be illustrated. The actual thrust in operation is obtained by using the recorded variables in the FDR, in particular the fuel flow rates.

The following climb equation (Lan & Roskam 2008) is to be satisfied in the least square sense over a 5-second internal:

$$\frac{dV}{g}\frac{dV}{dt} = T - D - W\sin\chi\tag{4.25}$$

and

134 Fuzzy Logic – Emerging Technologies and Applications

choice by correlating with a cosine wave with a constant frequency. The 20 points preceding and including the current time are employed in Eq. (4.20). The least-square method is found to converge well and gives reasonably accurate results. The lateral-directional equivalent

> *V*

*b V* 

where *c* is the mean chord length of wing airfoil section. The lateral-directional equivalent

As shown before, the thrust terms appear in the force equations and the pitching moment equation (Eqs. 4.12~4.14 and 4.16; but in the current application, Ty =Tz = 0.). Since the values of thrust for aircraft in flight cannot be directly measured in the current state of the art, they are not recorded in the FDR. The manufacturers of engines agreed that using such parameters as the Mach number, airspeed, flight altitude, temperature, the rpm of the pressure compressors and engine pressure ratios is adequate to estimate the engine thrust. A realistic thrust model is quite complex and cannot be represented by any simple equation. Since such thrust model is not available for the present study, a realistic one tied to the recorded engine performance parameters is developed with the fuzzy-logic algorithm.

For a commercial aircraft, most likely only the axial force and the pitching moment are affected by thrust. This assumption will be made in this Chapter. Theoretically, clear-air turbulence (i.e. random change in u, w (or ) and v (or )) affect the engine performance through its effects on static and dynamic distortions at the engine face. However, its effects are not known and cannot be estimated, and therefore ignored in the present application.

For the present purpose, data from the flight manual for the fuel flow rates ( *mf* ) at various altitudes (*h*), weights (*W*), Mach numbers (*M*), calibrated airspeed (CAS), engine pressure ratios (EPR), in cruise flight are utilized. Note that the drag polar for a given aircraft is generally not known to most researchers. To estimate it and hence the thrust magnitude in cruise, the assumption of a design lift-to-drag ratio (*L/D*) of 17.5 is made. This value of liftto-drag in cruise is assumed based on the past design experience for twin-jet transports. In the flight manual, various weights, altitudes, Mach numbers, CAS, EPR, and fuel flow rates in cruise are tabulated. The lift coefficient can be calculated at each flight condition immediately. As a result, the drag coefficient can be estimated from the assumption of liftto-drag ratio. Therefore, the design thrust in cruise at various Mach numbers can be estimated. For the Pratt & Whitney turbofan engines, thrust (*T*) is defined by EPR, so that

(4.21)

(4.22)

The local equivalent reduced frequency in the longitudinal motion is defined as,

reduced frequency is computed in the same manner.

 *k1*<sup>=</sup> *<sup>c</sup>*

 *k2*<sup>=</sup> <sup>2</sup>

reduced frequency is defined as

where *b* is the wing span.

**4.4 Fuzzy-Logic thrust model** 

the thrust model is set up as:

$$\frac{D}{dW} = \frac{D}{L}\cos\chi\tag{4.26}$$

All these equations are still valid in descent with negative climb angles (). The above equations are further employed for parameter identification in the process of modeling.

Once the thrust model is generated as a function of *h*, *W*, *M*, CAS, EPR, and *mf* with the flight conditions of climbing, cruise, and descent, one can estimate the thrust magnitude by inserting these flight variables from the FDR into the model.

#### **4.5 Fuzzy-Logic unsteady aerodynamic models**

Modeling means to establish the numerical relationship among certain variables of interest. In the fuzzy-logic model, more complete necessary influencing flight variables can be included to capture all possible effects on aircraft response to atmospheric disturbances. For longitudinal aerodynamics, the models are assumed to be of the form:

$$\mathbb{C}\_{\text{av}} \mathbb{C}\_{\text{av}} \mathbb{C}\_{\text{m}} = f\left(\alpha, \dot{\alpha}\,, q, k\_{\text{l}}, \beta, \delta\_{\text{v}} \,\, \middle|\, \delta\_{\text{v}} \,\, \middle|\, \text{M}, \, \eta, \, \delta\_{\text{v}} \,\, \middle|\, \text{T}\right) \tag{4.27}$$

where the left hand side represents the coefficients of axial force (*Cx*), normal force (*Cz*), and pitching moment (*Cm*), respectively. All variables on the right hand side of Eq. (4.27) have been defined in the previous section. It should be noted that the stabilizer angle (s) is included here, because it varies, though slowly, in flight to provide pitch trim (i.e. reducing the total static pitching moment to 0.0). The roll rate is included here because it is known that an aircraft under high aerodynamic loads at transonic speeds may have its longitudinal stability derivatives affected when additional disturbance due to roll rate is imposed.

For the lateral-directional aerodynamics,

Fig. 5. Predicted aerodynamic coefficients in normal force and moments for a twin-jet transport encountering severe atmospheric turbulence at cruise altitudes around 10,050 m

*P1*=2.61755+(1.26662)\*(0.79641)+(1.42338)\*(0.54764)+(2.07962)\*(0.70554)-

4478+(1.67592)\*(0.47678)+(1.13787)\*(0.3730)=11.04817

data of 0.81038, this prediction has an error of –0.88%.

becomes

cell to the total output is

11.04817\*1.08536E-004=1.19912E-003

around t = 3932 sec. Fig. 6(b) shows that

Other variable values are converted in the same way. It follows that the cell internal function

(0.44241)\*(0.03275)+(2.78017)\*(0.54475)+(1.7815)\*(0.66758)+(1.30818)\*(0.48299)+(1.82872)\*(0.3

The membership grades for the first cell are exactly equal to *xr*, being 0.79641, 0.54764, etc. Their product can be calculated to be 1.08536E-004. Therefore, the contribution of the first

The total output from all cells can be calculated to be 5.9962; while the denominator in Eq. (2.3) is calculated to be 7.46459. Therefore, the final prediction is 0.8033. Comparing with

To examine the stability characteristics, it is imperative to understand the flight environment in detail. The corresponding flight data are presented in Fig. 6. Note that *az* is the same as *an*, the normal acceleration. The variation of normal acceleration is presented in Fig. 6(a), showing the highest *an* being 1.75 g around t = 3930 sec and the lowest being 0.02 g

is approximately in phase with *an*. When *an* is the

$$\mathbf{C}\_{\mathcal{Y}}, \mathbf{C}\_{\boldsymbol{\theta}}, \mathbf{C}\_{\pi} \equiv \boldsymbol{f}\left(\boldsymbol{a}, \, \not\boldsymbol{\beta}, \, \not\boldsymbol{\phi}, \, \not\boldsymbol{\eta}, \, \not\boldsymbol{\xi}, \, \not\boldsymbol{\delta}, \, \not\boldsymbol{\delta}, \, \not\boldsymbol{\delta}, \, \not\boldsymbol{\delta}, \, \not\boldsymbol{\beta}\right) \tag{4.28}$$

where the left hand side represents the coefficients of side force (*Cy*), rolling moment (*Cl*) and yawing moment (*Cn*), respectively.

#### **4.6 Numerical results and discussions**

In the present study, the accuracy of the established unsteady aerodynamic models with six aerodynamic coefficients by using FLM technique is estimated by the sum of squared errors (SSE) and the square of multiple correlation coefficients (*R2*). Fig. 5 presents the aerodynamic coefficients of normal force *Cz*, pitching moment *Cm*, rolling moment *Cl*, and yawing moment *Cn* predicted by the unsteady aerodynamic models. The predicted data by the final refined models have good agreement with the flight data. The *Cm*-data scattering is most likely caused by turbulence-induced buffeting on the structure, in particular on the horizontal tail. Once the aerodynamic models are set up, one can calculate all necessary derivatives to analyze the stability.

The fuzzy-logic aerodynamic models are capable of generating the continuous derivatives for the static and dynamic stability study of a twin-jet transport in turbulence response. Firstly, how the fuzzy-logic prediction is achieved will be illustrated with one numerical example in the *Cz* calculation. At first, the range for each variable is defined to be larger than what actually occurred in the present set of *Cz*-data as follows:

[]=[-13,12], [ ]=[54,50], [*q*]=[-20,10], [*k1*]=[0,0.6], []=[-7,3], [*<sup>e</sup>*]=[-10,6], [*M*]=[0,1.6], [*p*]=[- 24,38], [*<sup>s</sup>*]=[-3,3], [ *q* ]= [4.964, 21.746]

For the first cell (1,1,1,1,1,1,1,1,1,1), the coefficients in Eq. (2.1) after model training are found to be:

1 *<sup>k</sup> p* =(2.61755, 1.26662, 1.42338, 2.07962, -0.44241, 2.78017, 1.78150, 1.30818, 1.82872, 1.67592, 1.13787).

Assume that in the following flight conditions *Cz* is to be predicted:

=6.91015 deg.; =2.95510 deg/sec; *q*=1.16609 deg/sec; *k1*= 0.01965; = -1.55252; *<sup>e</sup>* = 0.68120 deg; *M*=0.77279; *p*= -2.62359 deg/sec; *<sup>s</sup>*=-0.13930 deg, *q* =11.0545 kpa

These values of variables are converted to [0, 1]. For example,

*x*= [6.91015-(-13)]/[12-(-13)] = 0.79641

where the left hand side represents the coefficients of side force (*Cy*), rolling moment (*Cl*)

In the present study, the accuracy of the established unsteady aerodynamic models with six aerodynamic coefficients by using FLM technique is estimated by the sum of squared errors (SSE) and the square of multiple correlation coefficients (*R2*). Fig. 5 presents the aerodynamic coefficients of normal force *Cz*, pitching moment *Cm*, rolling moment *Cl*, and yawing moment *Cn* predicted by the unsteady aerodynamic models. The predicted data by the final refined models have good agreement with the flight data. The *Cm*-data scattering is most likely caused by turbulence-induced buffeting on the structure, in particular on the horizontal tail. Once the aerodynamic models are set up, one can calculate all necessary

The fuzzy-logic aerodynamic models are capable of generating the continuous derivatives for the static and dynamic stability study of a twin-jet transport in turbulence response. Firstly, how the fuzzy-logic prediction is achieved will be illustrated with one numerical example in the *Cz* calculation. At first, the range for each variable is defined to be larger than

For the first cell (1,1,1,1,1,1,1,1,1,1), the coefficients in Eq. (2.1) after model training are found

*<sup>k</sup> p* =(2.61755, 1.26662, 1.42338, 2.07962, -0.44241, 2.78017, 1.78150, 1.30818, 1.82872, 1.67592,

=2.95510 deg/sec; *q*=1.16609 deg/sec; *k1*= 0.01965;

]=[-7,3], [

*<sup>s</sup>*=-0.13930 deg, *q* =11.0545 kpa

*a* , *r*, *M*, , 

) (4.28)

*<sup>e</sup>*]=[-10,6], [*M*]=[0,1.6], [*p*]=[-

= -1.55252;

*<sup>e</sup>* =

, , , *p*, *r*, *k2*,

 *Cy*, *Cl*, *Cn*= *f* (

and yawing moment (*Cn*), respectively.

**4.6 Numerical results and discussions** 

derivatives to analyze the stability.

0.68120 deg; *M*=0.77279; *p*= -2.62359 deg/sec;

= [6.91015-(-13)]/[12-(-13)] = 0.79641

*<sup>s</sup>*]=[-3,3], [ *q* ]= [4.964, 21.746]

[

24,38], [

to be: 1

1.13787).

=6.91015 deg.;

*x*

]=[-13,12], [

what actually occurred in the present set of *Cz*-data as follows:

]=[54,50], [*q*]=[-20,10], [*k1*]=[0,0.6], [

Assume that in the following flight conditions *Cz* is to be predicted:

These values of variables are converted to [0, 1]. For example,

Fig. 5. Predicted aerodynamic coefficients in normal force and moments for a twin-jet transport encountering severe atmospheric turbulence at cruise altitudes around 10,050 m

Other variable values are converted in the same way. It follows that the cell internal function becomes

*P1*=2.61755+(1.26662)\*(0.79641)+(1.42338)\*(0.54764)+(2.07962)\*(0.70554)- (0.44241)\*(0.03275)+(2.78017)\*(0.54475)+(1.7815)\*(0.66758)+(1.30818)\*(0.48299)+(1.82872)\*(0.3 4478+(1.67592)\*(0.47678)+(1.13787)\*(0.3730)=11.04817

The membership grades for the first cell are exactly equal to *xr*, being 0.79641, 0.54764, etc. Their product can be calculated to be 1.08536E-004. Therefore, the contribution of the first cell to the total output is

11.04817\*1.08536E-004=1.19912E-003

The total output from all cells can be calculated to be 5.9962; while the denominator in Eq. (2.3) is calculated to be 7.46459. Therefore, the final prediction is 0.8033. Comparing with data of 0.81038, this prediction has an error of –0.88%.

To examine the stability characteristics, it is imperative to understand the flight environment in detail. The corresponding flight data are presented in Fig. 6. Note that *az* is the same as *an*, the normal acceleration. The variation of normal acceleration is presented in Fig. 6(a), showing the highest *an* being 1.75 g around t = 3930 sec and the lowest being 0.02 g around t = 3932 sec. Fig. 6(b) shows that is approximately in phase with *an*. When *an* is the

, ---) - *Cm* (

The roll damping (*Clp*) is extracted from the models of *Cl* with the central difference

 *Cl p* = [*Cl* (*---*, *p*+ *p*, *---*) - *Cl* (*---*, *p*- *p*, *---*)]/2*p* (4.30) where  *p* is in deg/sec. Similarly, all other aerodynamic derivatives are calculated by using

Before presenting the full aerodynamic characteristics, it is desirable to examine the effect of membership shape functions. The normal force coefficient, Cz = CN, and its derivatives in and d/dt play an important role in the plunging motion. Therefore, only these two derivatives are compared in Fig. 7. R2 for the triangular and parabolic shapes are 0.9787 and 0.9786, respectively. Although the values of R2 are close to each other, details in the

neighborhood of the peak values of the shape functions, the difference in the membership grades tends to be small. As a result, the effect of parabolic shape functions would smooth

> **3922 3924 3926 3928 3930 3932 3934 3936 t, sec.**

**triangular**

**3922 3924 3926 3928 3930 3932 3934 3936 t, sec.**

Fig. 7. Effects of membership shape functions on estimated - and d/dt- derivatives of CN

of a transport aircraft in atmospheric turbulence with plunging motion

**parabolic**


, ---)]/2

is perturbed by 0.5 degree while keeping all other

in plunging motion, probably because in the

(4.29)

 *Cm*

where

variables unchanged.

approach as follows:

the same method.

out the variation.

= [*Cm* ( +

=0.5 degree represents that

**4.6.1 Effects of membership shape functions** 

derivatives do differ, in particular in *CN*

**(a)**

**(b)**

**-250 -200 -150 -100 -50 0 50 100**

**C**

**N(d/dt), rad-1**

**C**

**N, rad-1**

highest (around *t* = 3930 sec), the aircraft rapidly plunging downward with the altitude (*h)* reaching the lowest as shown in Fig. 6(c); and is highest about 6.5 deg. in Fig. 6(b). At the same time, *M* is around 0.77 in Fig. 6(d). Since reaches a value about 6.5 deg in transonic flight, compressibility effect is important. It should be noted that the turbulent vertical wind field was not measured or estimated in the FDR; but is included in the total.

Fig. 6. The time history of flight variables for a twin-jet transport in severe atmospheric turbulence at the altitude around 10,050 m in transonic flight

The aerodynamic derivatives extracted from the unsteady aerodynamic models can be calculated with a central difference scheme. The longitudinal stability derivative (*Cm*) is extracted from the model of *Cm*. It is evaluated with the central difference approach as follows:

$$\mathbf{C}\_{ma} = \left[ \mathbf{C}\_{m} \left( a + \Lambda a \mathbf{z} \right) - \mathbf{\bar{c}}\_{m} \left( a \text{-} \Lambda a \mathbf{z} \text{--} \right) \right] / 2 \Delta a \tag{4.29}$$

where =0.5 degree represents that is perturbed by 0.5 degree while keeping all other variables unchanged.

The roll damping (*Clp*) is extracted from the models of *Cl* with the central difference approach as follows:

$$\mathbf{C}\_{l\cdot p} = [\mathbf{C}\_l \left( \begin{matrix} - \end{matrix}, p + \Delta \ p, \begin{matrix} - \end{matrix} \right) \cdot \mathbf{C}\_l \left( \begin{matrix} - \end{matrix}, p \cdot \Delta \ p, \begin{matrix} - \end{matrix} \right)] / 2\Delta p \tag{4.30}$$

where  *p* is in deg/sec. Similarly, all other aerodynamic derivatives are calculated by using the same method.

#### **4.6.1 Effects of membership shape functions**

138 Fuzzy Logic – Emerging Technologies and Applications

highest (around *t* = 3930 sec), the aircraft rapidly plunging downward with the altitude (*h)*

flight, compressibility effect is important. It should be noted that the turbulent vertical wind

Fig. 6. The time history of flight variables for a twin-jet transport in severe atmospheric

The aerodynamic derivatives extracted from the unsteady aerodynamic models can be calculated with a central difference scheme. The longitudinal stability derivative (*Cm*

extracted from the model of *Cm*. It is evaluated with the central difference approach as

turbulence at the altitude around 10,050 m in transonic flight

follows:

field was not measured or estimated in the FDR; but is included in the total

is highest about 6.5 deg. in Fig. 6(b). At the

reaches a value about 6.5 deg in transonic

.

> ) is

reaching the lowest as shown in Fig. 6(c); and

same time, *M* is around 0.77 in Fig. 6(d). Since

Before presenting the full aerodynamic characteristics, it is desirable to examine the effect of membership shape functions. The normal force coefficient, Cz = CN, and its derivatives in and d/dt play an important role in the plunging motion. Therefore, only these two derivatives are compared in Fig. 7. R2 for the triangular and parabolic shapes are 0.9787 and 0.9786, respectively. Although the values of R2 are close to each other, details in the derivatives do differ, in particular in *CN* in plunging motion, probably because in the neighborhood of the peak values of the shape functions, the difference in the membership grades tends to be small. As a result, the effect of parabolic shape functions would smooth out the variation.

Fig. 7. Effects of membership shape functions on estimated - and d/dt- derivatives of CN of a transport aircraft in atmospheric turbulence with plunging motion

Fig. 8. The time history of main longitudinal and lateral-directional of the static stability

Fig. 9 presents the time history of main longitudinal and lateral-directional oscillatory

 and 


During the plunging motion, the values have some differences between oscillatory and damping derivatives in Fig. 9(a) (C*mq* and (C*mq*)osc) and 9(c) (C*nr* and (C*nr*)osc) due to the effects

the directional characteristics more unstable (i.e. (*Cnr*)*osc* more positive). These results indicate that the turbulent crosswind has the effects on directional stability and damping. Although the dynamic derivatives tend to be small for the present configuration, these are much helpful to understand the unknown factors of instability characteristics. To be stable, (*Czq*)*osc* < 0, (*Cmq*)*osc* < 0, (*Clp*)*osc* < 0, and (*Cnr*)*osc* < 0. Physically, if it is unstable, the motion will


(4.31)

(4.32)

(4.33)

(4.34)




derivatives along the flight path

of the dynamic derivatives (i.e.

be divergent in oscillatory motions.

and 

derivatives along the flight path involving the

() *C CC mq osc m <sup>q</sup> <sup>m</sup>*

() *C CC zq osc z <sup>q</sup> <sup>z</sup>*

() *lp osc lp* sin *<sup>l</sup> C CC*

() cos *nr osc nr <sup>n</sup> C CC*

 and 


improve the stability in pitch after t = 3929.5 sec; while the effects of

In Fig. 9(c), the oscillatory derivatives are defined as

the oscillatory derivatives are defined as:

From a physical point of view, it is expected that in plunging motion the *CN* -derivative, which basically represents the virtual mass effect in unsteady aerodynamics (Sheu & Lan, 2011), should vary sharply. Note that the dynamic derivative, *CN* , is dimensionless (see below). In addition, with parabolic shape functions, modeling tends to take longer to converge. Therefore, in the following all derivatives are based on the model with the triangular membership shape functions.

#### **4.6.2 Stability derivatives for the whole time period**

The time period between 3927.5 sec and 3932.5 sec is emphasized in evaluating the stability characteristics, because of the plunging motion that affects the flight safety the most. All derivatives are converted to dimensionless ones in accordance with internationally known definition. For example, C*lp* is defined as C*l*/(pb/2V) and C*mq* as C*m*/(q *c* /2V), where *c* is the mean chord length. Therefore, the units of all aerodynamic derivatives are in rad-1. The main longitudinal and lateral-directional stability derivatives along the flight path are presented in Fig. 8. It should be noted that these derivatives are evaluated at the instantaneous conditions, instead of about the trim conditions as have been traditionally done. From the point of view in static stability, initially, the configuration has longitudinal stability (C*z* >0 and C*m* <0) as shown in Fig. 8(a), stable longitudinal damping (*Cmq* <0) in Fig. 8(b), lateral stability (*Cl* < 0) and directional stability (*Cn* > 0) in Fig. 8(c), small roll damping (*Clp* < 0) and insufficient directional damping (*Cnr* small or positive) in Fig. 8(d). During the plunging motion, in the period between t = 3928.5 sec. and t = 3930.5 sec, *Cm* > 0 and *Cl* > 0, so that the static stability becomes unstable. The aerodynamic instability is most likely caused by the motion that produces a time-dependent pressure distribution on the aircraft surface involving compressibility effects.

which basically represents the virtual mass effect in unsteady aerodynamics (Sheu & Lan,

below). In addition, with parabolic shape functions, modeling tends to take longer to converge. Therefore, in the following all derivatives are based on the model with the

The time period between 3927.5 sec and 3932.5 sec is emphasized in evaluating the stability characteristics, because of the plunging motion that affects the flight safety the most. All derivatives are converted to dimensionless ones in accordance with internationally known definition. For example, C*lp* is defined as C*l*/(pb/2V) and C*mq* as C*m*/(q *c* /2V), where *c* is the mean chord length. Therefore, the units of all aerodynamic derivatives are in rad-1. The main longitudinal and lateral-directional stability derivatives along the flight path are presented in Fig. 8. It should be noted that these derivatives are evaluated at the instantaneous conditions, instead of about the trim conditions as have been traditionally done. From the point of view in static stability, initially, the configuration has longitudinal

< 0) and directional stability (*Cn*

damping (*Clp* < 0) and insufficient directional damping (*Cnr* small or positive) in Fig. 8(d). During the plunging motion, in the period between t = 3928.5 sec. and t = 3930.5 sec, *Cm*

 > 0, so that the static stability becomes unstable. The aerodynamic instability is most likely caused by the motion that produces a time-dependent pressure distribution on the

<0) as shown in Fig. 8(a), stable longitudinal damping (*Cmq* <0) in

, is dimensionless (see

> 0) in Fig. 8(c), small roll

> 0


From a physical point of view, it is expected that in plunging motion the *CN*

2011), should vary sharply. Note that the dynamic derivative, *CN*

triangular membership shape functions.

stability (C*z*

and *Cl* Fig. 8(b), lateral stability (*Cl*

>0 and C*m*

aircraft surface involving compressibility effects.

**4.6.2 Stability derivatives for the whole time period** 

Fig. 8. The time history of main longitudinal and lateral-directional of the static stability derivatives along the flight path

Fig. 9 presents the time history of main longitudinal and lateral-directional oscillatory derivatives along the flight path involving the and -derivatives. Note that in Fig. 9(a), the oscillatory derivatives are defined as:

$$\left(\mathbf{C}\_{mq}\right)\_{\text{osc}} = \mathbf{C}\_{mq} + \mathbf{C}\_{m\dot{x}} \tag{4.31}$$

$$(\mathbf{C}\_{z\eta})\_{osc} = \mathbf{C}\_{z\eta} + \mathbf{C}\_{z\dot{\alpha}} \tag{4.32}$$

In Fig. 9(c), the oscillatory derivatives are defined as

$$\left(\mathbb{C}\_{lp}\right)\_{osc} = \mathbb{C}\_{lp} + \mathbb{C}\_{l\dot{\beta}} \sin \alpha \tag{4.33}$$

$$(\mathbf{C}\_{nr})\_{osc} = \mathbf{C}\_{nr} - \mathbf{C}\_{n\beta} \cos \alpha \tag{4.34}$$

During the plunging motion, the values have some differences between oscillatory and damping derivatives in Fig. 9(a) (C*mq* and (C*mq*)osc) and 9(c) (C*nr* and (C*nr*)osc) due to the effects of the dynamic derivatives (i.e. and -derivatives). The effects of -derivative on (*Czq*)*osc*, and -derivative on (*Clp*)*osc* are small. However, the effect of -derivative on (*Cmq*)*osc* is to improve the stability in pitch after t = 3929.5 sec; while the effects of -derivative is to cause the directional characteristics more unstable (i.e. (*Cnr*)*osc* more positive). These results indicate that the turbulent crosswind has the effects on directional stability and damping. Although the dynamic derivatives tend to be small for the present configuration, these are much helpful to understand the unknown factors of instability characteristics. To be stable, (*Czq*)*osc* < 0, (*Cmq*)*osc* < 0, (*Clp*)*osc* < 0, and (*Cnr*)*osc* < 0. Physically, if it is unstable, the motion will be divergent in oscillatory motions.

Eq. 3.4 for the example of an expression based on a linear theory), the following in-phase

2

0

2

0

After integration, Eq. (4.36) should produce Eq. (4.32) with Cz interpreted as CN. In addition, as shown in Fig. 10(a), the direction of the hysteretic curve is clockwise, and Eq. (4.36) should produce a positive value based on the linear theory. The sign of the integral (4.35), is represented by the slope of the hysteretic curve. Similarly, for the pitching moment, Fig. 10(b), the direction of the hysteretic curve is counterclockwise and hence the out-of-phase integral should produce a negative value according to the linear theory (i.e. stable damping). The example illustrates the fact that the present fuzzy-logic models can produce results to simulate the forced-oscillation testing. Typically, the linear results are used in

> **0 1 2 3 4 5 6 7 , deg.**

> **0 1 2 3 4 5 6 7 , deg.**

Fig. 10. Aerodynamic response due to a cosine harmonic oscillation at a reduced frequency

design; while the nonlinear results can be used in performance and simulation.

**(a)**

**(b)**

*C d <sup>N</sup>* 

 

*C d dd <sup>N</sup>* ( /)

 

 (4.35)

(4.36)

, q and *q* (see

change in α. According to a linear theory for CN and Cm as functions of α,

and out-of-phase integrals are given by: using CN as an example,

In-phase:

Out-of-phase:

**0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85**

**-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3**

of 0.02 in α as extracted from the fuzzy logic models.

**C**

**m**

**C**

**N**

Fig. 9. The time history of main longitudinal and lateral-directional oscillatory derivatives along the flight path

All derivatives in Eqs. (4.31) ~ (4.34) are estimated individually with the aerodynamic models and added afterwards to retain the nonlinearity. In wind-tunnel testing, these derivatives are not separately measured; instead they are determined in combination. As an example, assuming that it is desired to extract the response in CN and Cm at average conditions given by k1 = 0.02, β = -1.5, e = 0.0, M= 0.78, p = -3.0 deg/sec, s = -0.5, V=817 ft./sec., *q* =234 psf (see Eq. 4.27). The corresponding flight condition is approximately the one during the plunging motion. The angle of attack is assumed to vary harmonically (e.g. a cosine function) with a reduced frequency equal to k1. From the fuzzy-logic models, the response can be determined to be as shown in Fig. 10. The arrows represent the directions of

Fig. 9. The time history of main longitudinal and lateral-directional oscillatory derivatives

All derivatives in Eqs. (4.31) ~ (4.34) are estimated individually with the aerodynamic models and added afterwards to retain the nonlinearity. In wind-tunnel testing, these derivatives are not separately measured; instead they are determined in combination. As an example, assuming that it is desired to extract the response in CN and Cm at average conditions given by k1 = 0.02, β = -1.5, e = 0.0, M= 0.78, p = -3.0 deg/sec, s = -0.5, V=817 ft./sec., *q* =234 psf (see Eq. 4.27). The corresponding flight condition is approximately the one during the plunging motion. The angle of attack is assumed to vary harmonically (e.g. a cosine function) with a reduced frequency equal to k1. From the fuzzy-logic models, the response can be determined to be as shown in Fig. 10. The arrows represent the directions of

along the flight path

change in α. According to a linear theory for CN and Cm as functions of α, , q and *q* (see Eq. 3.4 for the example of an expression based on a linear theory), the following in-phase and out-of-phase integrals are given by: using CN as an example,

$$\text{In-phase: } \stackrel{2\pi}{\underset{0}{\text{In}}} \stackrel{2\pi}{\text{C}} \text{and} \theta \tag{4.35}$$

$$\text{Out-of-phase: } \stackrel{2\pi}{\underset{0}{\text{C}}} \mathbb{C}\_{N}(d\alpha \mid d\theta)d\theta \tag{4.36}$$

After integration, Eq. (4.36) should produce Eq. (4.32) with Cz interpreted as CN. In addition, as shown in Fig. 10(a), the direction of the hysteretic curve is clockwise, and Eq. (4.36) should produce a positive value based on the linear theory. The sign of the integral (4.35), is represented by the slope of the hysteretic curve. Similarly, for the pitching moment, Fig. 10(b), the direction of the hysteretic curve is counterclockwise and hence the out-of-phase integral should produce a negative value according to the linear theory (i.e. stable damping). The example illustrates the fact that the present fuzzy-logic models can produce results to simulate the forced-oscillation testing. Typically, the linear results are used in design; while the nonlinear results can be used in performance and simulation.

Fig. 10. Aerodynamic response due to a cosine harmonic oscillation at a reduced frequency of 0.02 in α as extracted from the fuzzy logic models.

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Roskam, J. (2003). Airplane Flight Dynamics and Automatic Flight Controls, Part I,

Sheu, D. & Lan, C. E. (2011). Estimation of Turbulent Vertical Velocity from Nonlinear Simulations of Aircraft Response, *Journal of Aircraft*, Vol. 48, No. 2, pp. 645-651. Wang, Z.; Lan, C. E. & Brandon, J. M. (1998). Fuzzy Logic Modeling of Nonlinear Unsteady

Wang, Z.; Lan, C. E. & Brandon, J. M. (1999). Fuzzy Logic Modeling of Lateral-Directional

Wang, Z.; Li, J.; Lan, C. E. & Brandon, J. M. (2001). Estimation of Unsteady Aerodynamic

Wang, Z.; Lan, C. E. & Brandon, J. M. (2002). Estimation of Lateral-Directional Unsteady

Weng, C. T. & Lan, C. E. (2008). Aerodynamic Analysis of a Landing Transport Airplane in Windshear, a monograph published by VDM Vertag Dr. Muller, Germany. Takagi, T. & Sugeno, M. (1985). Fuzzy identifications of systems and its applications to

Theodorsen, T. (1935). General Theory of Aerodynamic Instability and the Mechanism of

Institute of Aeronautics and Astronautics, Reston, Virginia, USA.

published by DAR Corporation, Lawrence, Kansas, USA.

*Aviation*, Series A, Vol. 38, No. 3, Sept. 2006, pp. 159-166.

2008, pp. 298-305.

45, No. 3, May–June 2008, pp. 916-922.

Lawrence, KS 66044, USA.

Space Administration, USA.

Company, Tucson, Arizona, USA.

Astronautics, Reston, Virginia, USA.

15, No. 1, pp. 116-132.

Hampton, Virginia, USA.

and Astronautics, Reston, Virginia, USA.

Aeronautics and Astronautics, Reston, Virginia, USA.

Reston, Virginia, USA.

Transport Aircraft Based on Flight Data," *Journal of Aeronautics, Astronautics and* 

on Transonic Lateral, Aerodynamics, *Journal of Aircraft*, Vol. 45, No. 1, Jan.-Feb.

Models for Identification of Uncommanded Rolling Motions, *Journal of Aircraft*, Vol.

AIAA paper 2003-6817, American Institute of Aeronautics and Astronautics,

and Control, NASA Reference Publication No. 1168, National Aeronautics and

Aerodynamics, AIAA Paper 98-4351, American Institute of Aeronautics and

Unsteady Aerodynamics, AIAA Paper 99-4012, American Institute of Aeronautics

Models from Flight Test Data," AIAA Paper 2001-4017, American Institute of

Aerodynamic Models from Flight Test Data, AIAA Paper 2002-4626, American

modeling and control, *IEEE Transactions on Systems, Man and Cybernetics*, Vol. SMC-

Flutter, NACA Report 496, National Advisory Committee for Aeronautics,

#### **4.7 Flight dynamic application**

As indicated in Introduction, the aerodynamic models generated by the FLM algorithm can serve as the forcing functions and be coupled with the dynamic equations of motion for flight simulation or flight reconstruction in accident investigation. However, it was found that the flight dynamic equations require reformulation to improve numerical damping and avoid numerical divergence (Sheu & Lan 2011). As a result of numerical integration, the turbulent vertical wind can also be estimated from the difference in the total α as measured by the aircraft α-sensor and the motion-produced α by numerical integration. The numerical example presented in the quoted reference is based on the same flight data examined in this Section.

#### **5. Conclusions**

The main objective in this paper was to illustrate the nonlinear unsteady aerodynamic models based on the FLM technique having the capability to evaluate the variations in stability of commercial aircraft with adverse weather effects. The present FLM technique was explained in detail and verified with simple examples and wind-tunnel data. It was shown that the FLM technique was capable of handling nonlinear and unsteady aerodynamic environment exhibited for a twin-jet transport in severe atmospheric turbulence with sudden plunging motion in transonic flight. The predicted results showed that the models could produce reasonable aerodynamic coefficients and several derivatives for the assessment of stability characteristics, especially for the study of unknown factors in adverse weather conditions.

At the present time, any aircraft encountering severe atmospheric turbulence is considered uncontrollable. Since the aerodynamics represented by the fuzzy-logic models is realistic, they can be coupled with the numerical integration of flight dynamic equations to study possible improvement in controllability. However, to develop the control law, it is imperative to include the unsteady and nonlinear aerodynamic effects.

#### **6. References**


As indicated in Introduction, the aerodynamic models generated by the FLM algorithm can serve as the forcing functions and be coupled with the dynamic equations of motion for flight simulation or flight reconstruction in accident investigation. However, it was found that the flight dynamic equations require reformulation to improve numerical damping and avoid numerical divergence (Sheu & Lan 2011). As a result of numerical integration, the turbulent vertical wind can also be estimated from the difference in the total α as measured by the aircraft α-sensor and the motion-produced α by numerical integration. The numerical example presented in the quoted reference is based on the same flight data examined in this

The main objective in this paper was to illustrate the nonlinear unsteady aerodynamic models based on the FLM technique having the capability to evaluate the variations in stability of commercial aircraft with adverse weather effects. The present FLM technique was explained in detail and verified with simple examples and wind-tunnel data. It was shown that the FLM technique was capable of handling nonlinear and unsteady aerodynamic environment exhibited for a twin-jet transport in severe atmospheric turbulence with sudden plunging motion in transonic flight. The predicted results showed that the models could produce reasonable aerodynamic coefficients and several derivatives for the assessment of stability characteristics, especially for the study of unknown factors in

At the present time, any aircraft encountering severe atmospheric turbulence is considered uncontrollable. Since the aerodynamics represented by the fuzzy-logic models is realistic, they can be coupled with the numerical integration of flight dynamic equations to study possible improvement in controllability. However, to develop the control law, it is

Chang, R. C.; Ye, C. E.; Lan, C. E. & Guan, W. L. (2009). Flying Qualities for a Twin-Jet

Gelb, A. (1982). *Applied Optimal Estimation*, The M. I. T. Press, Cambridge, Massachusetts,

Klein, V.; Batterson, J. G. & Murphy, P. C. (1981), Determination of Airplane Model

Lan C E & Guan M., (2005). Flight Dynamic Analysis of a Turboprop Transport Airplane in

Transport in Severe Atmospheric Turbulence, *AIAA Journal of Aircraft*, Vol. 46, No.

Structure from Flight Data by Using Modified Stepwise Regression, NASA Technical Publication No. 1916, National Aeronautics and Space Administration,

Icing Accident, AIAA Paper 2005-5922, American Institute of Aeronautics and

imperative to include the unsteady and nonlinear aerodynamic effects.

**4.7 Flight dynamic application** 

Section.

**5. Conclusions** 

adverse weather conditions.

5, pp. 1673-1680.

Astronautics, Reston, Virginia, USA.

**6. References** 

USA.

USA.


**0**

**8**

*Pakistan*

**Adaptive Fuzzy Wavelet NN Control Strategy for**

In the last few years, different linear and non-linear control techniques have been applied by many researchers on the vehicle suspension system. The basic purpose of suspension system is to improve the ride comfort and better road handling capability. Therefore, a comfortable and fully controlled ride can not be guaranteed without a good suspension system. The

The passive suspension system is an open loop control system consisting of the energy storing (spring) and dissipating element (damper). The passive suspension performance depends on the road profile, controlling the relative movement of the body and tires by using various kinds of damping and energy dissipating elements. Passive suspension has considerable restriction in structural applications. The features are resolved by the designers with respect to the design objectives and the proposed application. All the ongoing research in this area

• minimize the effect of road and inertial disturbances, on human body, caused by cornering

All the above objectives lead to rapidly changing operating conditions and the passive suspension system is not as efficient to cope with them by adapting its parameters, simultaneously. So, there would always be a compromise between comfort and safety for

Semi-active suspension system consists of a sensor that identifies bumps on the road and motion of the vehicle and a controller that controls the damper on each wheel. The semi-active suspension can respond to even small variations in road area and cornering. It offers quick variations in rate of springs damping coefficients. This suspension system does not give any energy to the system but damper is changed by the controller. The controller resolves the rank of damping based on control approach and automatically changes the damper according to the preferred levels. Actuator and sensors are attached to sense the road profile for the control input. The adaptive fuzzy controller for semi-active suspension systems was presented by

• good control on all the four wheels of the car for their optimal contact with road.

suspension system can be categorized as; Passive, Semi-active and Active.

mainly caters the following issues to improve the suspension control;

• minimize the vertical car body displacement and acceleration.

**1. Introduction**

or braking.

passive suspension system.

**Full Car Suspension System**

Laiq Khan, Rabiah Badar and Shahid Qamar

*COMSATS Institute of Information Technology, Abbottabad*

*Department of Electrical Engineering,*

Zadeh, L. A. (1973). Outline of a New Approach to the Analysis of Complex Systems and Decision Processes, *IEEE Transactions on Systems, Man, and Cybernetics*, Vol. SMC-3, No. 1, pp. 28-44.

### **Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System**

Laiq Khan, Rabiah Badar and Shahid Qamar *Department of Electrical Engineering, COMSATS Institute of Information Technology, Abbottabad*

#### *Pakistan*

#### **1. Introduction**

146 Fuzzy Logic – Emerging Technologies and Applications

Zadeh, L. A. (1973). Outline of a New Approach to the Analysis of Complex Systems and

No. 1, pp. 28-44.

Decision Processes, *IEEE Transactions on Systems, Man, and Cybernetics*, Vol. SMC-3,

In the last few years, different linear and non-linear control techniques have been applied by many researchers on the vehicle suspension system. The basic purpose of suspension system is to improve the ride comfort and better road handling capability. Therefore, a comfortable and fully controlled ride can not be guaranteed without a good suspension system. The suspension system can be categorized as; Passive, Semi-active and Active.

The passive suspension system is an open loop control system consisting of the energy storing (spring) and dissipating element (damper). The passive suspension performance depends on the road profile, controlling the relative movement of the body and tires by using various kinds of damping and energy dissipating elements. Passive suspension has considerable restriction in structural applications. The features are resolved by the designers with respect to the design objectives and the proposed application. All the ongoing research in this area mainly caters the following issues to improve the suspension control;


All the above objectives lead to rapidly changing operating conditions and the passive suspension system is not as efficient to cope with them by adapting its parameters, simultaneously. So, there would always be a compromise between comfort and safety for passive suspension system.

Semi-active suspension system consists of a sensor that identifies bumps on the road and motion of the vehicle and a controller that controls the damper on each wheel. The semi-active suspension can respond to even small variations in road area and cornering. It offers quick variations in rate of springs damping coefficients. This suspension system does not give any energy to the system but damper is changed by the controller. The controller resolves the rank of damping based on control approach and automatically changes the damper according to the preferred levels. Actuator and sensors are attached to sense the road profile for the control input. The adaptive fuzzy controller for semi-active suspension systems was presented by

controller to control the active suspension system. The fuzzy control for active suspension system presented by (Yester & Jr., 1992) considers only the ride comfort. (Rao & Prahlad, 1997) proposed a tuneable fuzzy logic controller, on active suspension system without taking into account the nonlinear features of the suspension spring and shock absorber, also, the robustness problem was not discussed. The neural network control system applied on active suspension system has been discussed by (Moran & Masao, 1994) but does not give enough information about the robustness and sensitivity properties of the neural control towards the parameter deviations and model uncertainties. Also, sliding mode neural network inference fuzzy logic control for active suspension systems is presented by (Al-Holou et al., 2002), but did not give any information about the rattle space limits. (Huang & Lin, 2003; Lin & Lian, 2008) proposed a DSP-based self-organizing fuzzy controller for an active suspension system of car, to reduce the displacement and acceleration in the sprung mass so as to improve the handling performance and ride comfort of the car. (Lian et al., Feb. 2005) proposed a fuzzy

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 149

However, it is still complicated to design suitable membership functions and fuzzy linguistic rules of the fuzzy logic controllers to give suitable learning rate and weighting distribution

Since, the aforementioned fuzzy logic and neural network controllers on active models, did not give enough information about the robustness, sensitivity and rattle space limits. These techniques were combined with wavelets to solve different control and signal processing problems and collectively known as Fuzzy Wavelet Neural Networks (FWNNs) (Chalasani, 1986; Hac, 1986; Heo et al., 2000; Meld, 1991; Thompson & Davis, 2005; Thompson & Pearce, 1998). The combination of a fuzzy wavelet neural inference system comprises the strength of the optimal definitions of the antecedent part and the consequent part of the fuzzy rules. In this study, fuzzy wavelet neural network control is proposed for the active suspension control. A FWNN combines wavelet theory with fuzzy logic and neural networks. Wavelet neural networks are based on wavelet transform which has the capability to examine non-stationary signals to determine their local details. Fuzzy logic system decreases the complexity and deals with vagueness of the data. Neural networks have self-learning qualities that raises the precision of the model. Their arrangement permits to build up a system with fast learning abilities that can explain nonlinear structures. Different structures of FWNN have been proposed in the literature. Due to its strong estimation and controlling properties FWNN has found extensive applications in the areas of identification and control of non-linear plants (Abiyev & Kaynak, 2008; Adeli & Jiang, 2006; Banakar & Azeem, 2008; Yilmaz & Oysal, 2010). In this chapter, different softcomputing techniques have been combined with wavelets for the active suspension control of full car model to minimize the vibrations of the vehicle against the road disturbances. The proposed Adaptive Fuzzy Wavelet Neural Network (AFWNN) control integrates the ability of wavelet to analyze the local details of the signal with that of fuzzy logic to reduce system complexity and with the self learning capability of neural networks, which makes the controller efficient for controlling unknown dynamic plants. The results of the proposed models have been compared with passive and semi-active suspension system. The robustness of the system has further been evaluated by comparing the results

This chapter has been arranged as follows; Section2 gives the structural and mathematical details of the proposed AFWNN models. In Section 3 the modeling details and closed loop

controller to control the active suspension system.

parameters in the self-organizing fuzzy controller.

with Adaptive PID (APID).

(Lieh & Li, 1997) which shows only the acceleration of the vehicle compared to the passive suspension.

On the other hand, active suspension consists of actuator. The controller drives the actuator, which depends on the proposed control law. The active suspension system gives the freedom to tune the whole suspension system and the control force can be initiated locally or globally depending on the system state. The active suspension systems provide more design flexibility and increase the range of achievable objectives. The active suspension passenger seat is proposed by (Stein & Ballo, 1991) for off-road vehicles. Also, the passenger suspension seat was considered by (Nicolas et al., 1997) in their control technique to improve ride comfort. Various control techniques such as optimal state-feedback (Esmailzadeh & Taghirad, 1996), model reference adaptive control (Sunwoo et al., June 1991), backstepping method (Lin & Kanellakopoulos, 1997), fuzzy control (Yoshimura et al., 1999) and sliding mode control (Yoshimura et al., 2001) have been presented in the last few years for optimized control of the active suspension system.

In order to examine these suspension systems, three types of car model have been introduced in the literature; Quarter car model, Half car model and Full car model. In car modeling, quarter car model is the simplest one. Many approaches on quarter car suspension systems have been carried out by (Hac, 1987; Yue et al., 1988) but do not reveal robustness of the system. The robustness of quarter-car suspension system based on stochastic stability has been presented by (Ray, 1991) but this technique needs large feedback gains and an appropriate phase must be chosen. The best performance estimations of variable suspension system on a quarter car model are observed by (Redfield & Karnopp, 1988). Various linear control techniques are applied on a quarter car model in (Bigarbegian et al., 2008) but did not give any information for large gain from road disturbance to vehicle body acceleration. The dynamic behavior and vibration control of a half-car suspension model is inspected by different researchers in (Hac, 1986; Krtolica & Hrovat, 1990; 1992; Thompson & Davis, 2005; Thompson & Pearce, 1998).

The active control of seat for full car model is examined by (Rahmi, 2003). Some control approaches have been examined to minimize the vertical motion, roll and also the chassis motion of vehicle by (Barak & Hrovat, 1988; Cech, 1994; Crolla & Abdel−Hady, 1991). The PID controller is applied on active suspension system by (Kumar, 2008). The combined H∞ controller with LQR controller on an active car suspension is given by (Kaleemullah et al., 2011), but this controller requires the frequency characterization of the system uncertainties and plant disturbance, which are usually not available. An experimental 1-DOF microcomputerized based suspension system was presented by (White-Smoke, 2011), using actuator force as control input. However, the extension of this model to other practical models is not straightforward.

Fuzzy logic control has been utilized widely for the control applications. Such a control approach has the definite characteristic of being able to build up the controller without mathematical model of the system. Therefore, it has been employed to control active suspension systems (Hedrick & Butsuen, 1990; Hrovat, 1982; Meller, 1978; Smith, 1995).

In (Nicolas et al., 1997), the authors used a fuzzy logic controller to increase the ride comfort of the vehicle. A variety of simulations showed that the fuzzy logic control is proficient to give a better ride quality than other common control approaches for example, skyhook control (Ahmadian & Pare, 2000; Bigarbegian et al., 2008). (Lian et al., Feb. 2005) proposed a fuzzy 2 Will-be-set-by-IN-TECH

(Lieh & Li, 1997) which shows only the acceleration of the vehicle compared to the passive

On the other hand, active suspension consists of actuator. The controller drives the actuator, which depends on the proposed control law. The active suspension system gives the freedom to tune the whole suspension system and the control force can be initiated locally or globally depending on the system state. The active suspension systems provide more design flexibility and increase the range of achievable objectives. The active suspension passenger seat is proposed by (Stein & Ballo, 1991) for off-road vehicles. Also, the passenger suspension seat was considered by (Nicolas et al., 1997) in their control technique to improve ride comfort. Various control techniques such as optimal state-feedback (Esmailzadeh & Taghirad, 1996), model reference adaptive control (Sunwoo et al., June 1991), backstepping method (Lin & Kanellakopoulos, 1997), fuzzy control (Yoshimura et al., 1999) and sliding mode control (Yoshimura et al., 2001) have been presented in the last few years for optimized control of

In order to examine these suspension systems, three types of car model have been introduced in the literature; Quarter car model, Half car model and Full car model. In car modeling, quarter car model is the simplest one. Many approaches on quarter car suspension systems have been carried out by (Hac, 1987; Yue et al., 1988) but do not reveal robustness of the system. The robustness of quarter-car suspension system based on stochastic stability has been presented by (Ray, 1991) but this technique needs large feedback gains and an appropriate phase must be chosen. The best performance estimations of variable suspension system on a quarter car model are observed by (Redfield & Karnopp, 1988). Various linear control techniques are applied on a quarter car model in (Bigarbegian et al., 2008) but did not give any information for large gain from road disturbance to vehicle body acceleration. The dynamic behavior and vibration control of a half-car suspension model is inspected by different researchers in (Hac, 1986; Krtolica & Hrovat, 1990; 1992; Thompson & Davis, 2005;

The active control of seat for full car model is examined by (Rahmi, 2003). Some control approaches have been examined to minimize the vertical motion, roll and also the chassis motion of vehicle by (Barak & Hrovat, 1988; Cech, 1994; Crolla & Abdel−Hady, 1991). The PID controller is applied on active suspension system by (Kumar, 2008). The combined H∞ controller with LQR controller on an active car suspension is given by (Kaleemullah et al., 2011), but this controller requires the frequency characterization of the system uncertainties and plant disturbance, which are usually not available. An experimental 1-DOF microcomputerized based suspension system was presented by (White-Smoke, 2011), using actuator force as control input. However, the extension of this model to other practical models

Fuzzy logic control has been utilized widely for the control applications. Such a control approach has the definite characteristic of being able to build up the controller without mathematical model of the system. Therefore, it has been employed to control active suspension systems (Hedrick & Butsuen, 1990; Hrovat, 1982; Meller, 1978; Smith, 1995).

In (Nicolas et al., 1997), the authors used a fuzzy logic controller to increase the ride comfort of the vehicle. A variety of simulations showed that the fuzzy logic control is proficient to give a better ride quality than other common control approaches for example, skyhook control (Ahmadian & Pare, 2000; Bigarbegian et al., 2008). (Lian et al., Feb. 2005) proposed a fuzzy

suspension.

the active suspension system.

Thompson & Pearce, 1998).

is not straightforward.

controller to control the active suspension system. The fuzzy control for active suspension system presented by (Yester & Jr., 1992) considers only the ride comfort. (Rao & Prahlad, 1997) proposed a tuneable fuzzy logic controller, on active suspension system without taking into account the nonlinear features of the suspension spring and shock absorber, also, the robustness problem was not discussed. The neural network control system applied on active suspension system has been discussed by (Moran & Masao, 1994) but does not give enough information about the robustness and sensitivity properties of the neural control towards the parameter deviations and model uncertainties. Also, sliding mode neural network inference fuzzy logic control for active suspension systems is presented by (Al-Holou et al., 2002), but did not give any information about the rattle space limits. (Huang & Lin, 2003; Lin & Lian, 2008) proposed a DSP-based self-organizing fuzzy controller for an active suspension system of car, to reduce the displacement and acceleration in the sprung mass so as to improve the handling performance and ride comfort of the car. (Lian et al., Feb. 2005) proposed a fuzzy controller to control the active suspension system.

However, it is still complicated to design suitable membership functions and fuzzy linguistic rules of the fuzzy logic controllers to give suitable learning rate and weighting distribution parameters in the self-organizing fuzzy controller.

Since, the aforementioned fuzzy logic and neural network controllers on active models, did not give enough information about the robustness, sensitivity and rattle space limits. These techniques were combined with wavelets to solve different control and signal processing problems and collectively known as Fuzzy Wavelet Neural Networks (FWNNs) (Chalasani, 1986; Hac, 1986; Heo et al., 2000; Meld, 1991; Thompson & Davis, 2005; Thompson & Pearce, 1998). The combination of a fuzzy wavelet neural inference system comprises the strength of the optimal definitions of the antecedent part and the consequent part of the fuzzy rules. In this study, fuzzy wavelet neural network control is proposed for the active suspension control. A FWNN combines wavelet theory with fuzzy logic and neural networks. Wavelet neural networks are based on wavelet transform which has the capability to examine non-stationary signals to determine their local details. Fuzzy logic system decreases the complexity and deals with vagueness of the data. Neural networks have self-learning qualities that raises the precision of the model. Their arrangement permits to build up a system with fast learning abilities that can explain nonlinear structures. Different structures of FWNN have been proposed in the literature. Due to its strong estimation and controlling properties FWNN has found extensive applications in the areas of identification and control of non-linear plants (Abiyev & Kaynak, 2008; Adeli & Jiang, 2006; Banakar & Azeem, 2008; Yilmaz & Oysal, 2010).

In this chapter, different softcomputing techniques have been combined with wavelets for the active suspension control of full car model to minimize the vibrations of the vehicle against the road disturbances. The proposed Adaptive Fuzzy Wavelet Neural Network (AFWNN) control integrates the ability of wavelet to analyze the local details of the signal with that of fuzzy logic to reduce system complexity and with the self learning capability of neural networks, which makes the controller efficient for controlling unknown dynamic plants. The results of the proposed models have been compared with passive and semi-active suspension system. The robustness of the system has further been evaluated by comparing the results with Adaptive PID (APID).

This chapter has been arranged as follows; Section2 gives the structural and mathematical details of the proposed AFWNN models. In Section 3 the modeling details and closed loop

In a TSK fuzzy model, each rule is divided into two regions, represented by the IF-THEN statement. In the IF part of the fuzzy rule, membership functions are given, and in the THEN part of the fuzzy rule a linear function of inputs or a constant is used. These rules are based on either experts knowledge or adaptive learning. The wavelets can collect the information globally and locally easily by means of the multiresolution property (Ho et al., 2001). The

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 151

Where, *x*1, *x*2, ..., *xm*, *y*1, *y*2, ..., *yn* are the input-output variables and *Aij* is the membership function of *ith* input and *jth* rule. Wavelet functions are in the consequent part of the rules. The entire fuzzy model can be attained by finding/learning the parameters of antecedent and

The AFWNN structure has been depicted in Figure 1. This structure comprises of combination of the two network structures, i.e., upper side and lower side. Where, upper side encloses

**Layer 1:** This is the first layer of fuzzy reasoning as well as the wavelet network. This layer

**Layer 2:** In this layer fuzzification process is performed and neurons represent fuzzy sets used in the antecedent part of the linguistic fuzzy rules. The outputs of this layer are the values of

**Layer 3:** In this layer each node represents a fuzzy rule. In order to compute the firing strength

*i*

**Layer 4:** In this layer, wavelet functions are represented. The output of this layer is given by;

*qil* = *f*(*xi*, *bil*, *ail*)

is the min operation and *μj*(*x*) are the input values for the next layer (consequent

of each rule, and min operation is used to estimate the output value of the layer. i.e.,

*μj*(*x*) = ∏

*m* ∑ *i*=1

*m* ∑ *i*=1

*m* ∑ *i*=1

. . . *wi*1(<sup>1</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup>

*wi*2(<sup>1</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup>

*win*(<sup>1</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup>

*ηj*(*xi*) (3)

*yl* = *wlψl*(*q*) (4)

*<sup>i</sup>*1)*e* − *q*2 *i*1 2

*<sup>i</sup>*2)*e* − *q*2 *i*2 2

*in*)*e* − *q*2 *in* 2

proposed AFWNN model has fast convergence and accuracy properties.

*If x*<sup>1</sup> *isA*<sup>11</sup> *and x*<sup>2</sup> *is A*<sup>12</sup> *and* ... *xm is A*1*mThen y*<sup>1</sup> =

*If x*<sup>1</sup> *isA*<sup>21</sup> *and x*<sup>2</sup> *is A*<sup>22</sup> *and* ... *xm is A*2*mThen y*<sup>2</sup> =

*If x*<sup>1</sup> *isAn*<sup>1</sup> *and x*<sup>2</sup> *is An*<sup>2</sup> *and* ... *xm is AnmThen yn* =

wavelet neural network and lower side encloses fuzzy reasoning process.

accepts input values. Its nodes transmit input values to the next layer.

The whole network works in a layered fashion, as follows;

The AFWNN rules have the following form;

consequent part.

where, ∏ *i*

layer).

the membership functions '*ηj*(*xj*)'.

Where, *ψ<sup>l</sup>* = *f*(*ail*, *qil*) is a functional such that

system have been discussed. Section 4 gives the simulation results and discussion. Finally, section 5 concludes our work.

#### **2. Fuzzy wavelet neural network control**

Wavelet neural network is a new and innovative network, which is based on wavelet transforms (Oussar & Dreyfus, 2000). The structural design of the wavelet neural network is laid on a multilayered perceptron. A discrete wavelet function is applied as node activation function in the wavelet neural network. Because, the wavelet space is utilized as a feature space of pattern identification, the feature extraction of signal is recognized by the weighted sum of the inner product of wavelet base and signal vector. Furthermore, network acquires the ability of approximation and robustness. The entire estimation is on the logistic infrastructure. Wavelets can be expressed as follows:

$$\Psi\_j(\mathbf{x}) = |a\_j|^{\frac{-1}{2}} \Psi\left(\frac{\mathbf{x} - b\_j}{a\_j}\right), \quad a\_j \neq 0, \quad j = 1, 2, \dots, n \tag{1}$$

Where, Ψ*j*(*x*) is the family of wavelets, *x* = *x*1, *x*2, ..., *xm* shows the input values, *aj* = *a*1*j*, *a*2*j*, ..., *amj* and *bj* = *b*1*j*, *b*2*j*, ..., *bmj* represent the dilation and translation parameters of the mother wavelet Ψ(*x*), respectively. The Ψ(*x*) function is a waveform of limited duration and has a zero mean value.

Wavelet neural networks are mainly three layered networks using wavelets as activation function. The output for wavelet neural network is formulated as;

$$y = \sum\_{j=1}^{k} w\_j \Psi\_j(\mathbf{x}) \tag{2}$$

Where, Ψ*j*(*x*) is the wavelet function of the *jth* part of hidden layer, because, the wavelet networks contain the wavelet functions in the hidden layer's neurons of the network. *wj* are the weights connected between the hidden layer and the output layer.

Wavelet functions have capability of time−frequency localization property (Zhang & Benveniste, 1992). Localization of the *ith* hidden layer of wavelet neural network is found by the dilation and translation parameters of the wavelet function. The dilation parameter controls the spread of the wavelet and the translation parameter determines the center position of the wavelet (Y. Chen & Dong, 2006).

Normally, two techniques are used for signifying multidimensional wavelets. In the first technique, they are created by using the product of one-dimensional wavelet functions. This wavelet neural network technique model is used by (Zhang et al., 1995). In second technique, the Euclidian norms of the input variables are used as the inputs of one-dimensional wavelets (Billings & Wei, 2005; Zhang, 1997).

The proposed AFWNN incorporates wavelet functions in the conventional TSK fuzzy logic system. In the conventional approach, a linear function or constant is used in the consequent part of the linguistic rules for TSK fuzzy system. In the AFWNN, wavelet functions are used in the consequent part to enhance the estimation capability and computational strength of the neuro-fuzzy system by utilizing their time-frequency localization property.

In a TSK fuzzy model, each rule is divided into two regions, represented by the IF-THEN statement. In the IF part of the fuzzy rule, membership functions are given, and in the THEN part of the fuzzy rule a linear function of inputs or a constant is used. These rules are based on either experts knowledge or adaptive learning. The wavelets can collect the information globally and locally easily by means of the multiresolution property (Ho et al., 2001). The proposed AFWNN model has fast convergence and accuracy properties.

The AFWNN rules have the following form;

4 Will-be-set-by-IN-TECH

system have been discussed. Section 4 gives the simulation results and discussion. Finally,

Wavelet neural network is a new and innovative network, which is based on wavelet transforms (Oussar & Dreyfus, 2000). The structural design of the wavelet neural network is laid on a multilayered perceptron. A discrete wavelet function is applied as node activation function in the wavelet neural network. Because, the wavelet space is utilized as a feature space of pattern identification, the feature extraction of signal is recognized by the weighted sum of the inner product of wavelet base and signal vector. Furthermore, network acquires the ability of approximation and robustness. The entire estimation is on the logistic infrastructure.

Where, Ψ*j*(*x*) is the family of wavelets, *x* = *x*1, *x*2, ..., *xm* shows the input values, *aj* = *a*1*j*, *a*2*j*, ..., *amj* and *bj* = *b*1*j*, *b*2*j*, ..., *bmj* represent the dilation and translation parameters of the mother wavelet Ψ(*x*), respectively. The Ψ(*x*) function is a waveform of limited duration and

Wavelet neural networks are mainly three layered networks using wavelets as activation

Where, Ψ*j*(*x*) is the wavelet function of the *jth* part of hidden layer, because, the wavelet networks contain the wavelet functions in the hidden layer's neurons of the network. *wj* are

Wavelet functions have capability of time−frequency localization property (Zhang & Benveniste, 1992). Localization of the *ith* hidden layer of wavelet neural network is found by the dilation and translation parameters of the wavelet function. The dilation parameter controls the spread of the wavelet and the translation parameter determines the center

Normally, two techniques are used for signifying multidimensional wavelets. In the first technique, they are created by using the product of one-dimensional wavelet functions. This wavelet neural network technique model is used by (Zhang et al., 1995). In second technique, the Euclidian norms of the input variables are used as the inputs of one-dimensional wavelets

The proposed AFWNN incorporates wavelet functions in the conventional TSK fuzzy logic system. In the conventional approach, a linear function or constant is used in the consequent part of the linguistic rules for TSK fuzzy system. In the AFWNN, wavelet functions are used in the consequent part to enhance the estimation capability and computational strength of the

neuro-fuzzy system by utilizing their time-frequency localization property.

, *aj* � 0, *j* = 1, 2, ..., *n* (1)

*wj*Ψ*j*(*x*) (2)

section 5 concludes our work.

**2. Fuzzy wavelet neural network control**

Wavelets can be expressed as follows:

has a zero mean value.

Ψ*j*(*x*) = |*aj*|

−1 <sup>2</sup> Ψ

function. The output for wavelet neural network is formulated as;

the weights connected between the hidden layer and the output layer.

position of the wavelet (Y. Chen & Dong, 2006).

(Billings & Wei, 2005; Zhang, 1997).

 *<sup>x</sup>* <sup>−</sup> *bj aj*

> *y* = *k* ∑ *j*=1

$$If \ x\_1 \text{ is} A\_{11} \text{ and } \mathbf{x\_2} \text{ is } A\_{12} \text{ and } \dots \text{ x\_m \text{ is } A\_{1m}} Then \ y\_1 = \sum\_{i=1}^m w\_{i1} (1 - q\_{i1}^2) e^{-\frac{q\_{i1}^2}{2}}$$

$$If \ x\_1 \text{ is} A\_{21} \text{ and } \mathbf{x\_2} \text{ is } A\_{22} \text{ and } \dots \text{ x\_m \text{ is } A\_{2m}} Then \ y\_2 = \sum\_{i=1}^m w\_{i2} (1 - q\_{i2}^2) e^{-\frac{q\_{i2}^2}{2}}$$

$$\begin{array}{c} \vdots\\ \vdots\\ \text{If } \mathbf{x}\_1 \text{ is} \mathbf{A}\_{n1} \text{ and } \mathbf{x}\_2 \text{ is } \mathbf{A}\_{n2} \text{ and } \dots \text{ x}\_m \text{ is } \mathbf{A}\_{nm}\\ \text{Then } y\_n = \sum\_{i=1}^m w\_{in} (1 - q\_{in}^2) e^{-\frac{q\_{in}^2}{2}} \end{array}$$

Where, *x*1, *x*2, ..., *xm*, *y*1, *y*2, ..., *yn* are the input-output variables and *Aij* is the membership function of *ith* input and *jth* rule. Wavelet functions are in the consequent part of the rules. The entire fuzzy model can be attained by finding/learning the parameters of antecedent and consequent part.

The AFWNN structure has been depicted in Figure 1. This structure comprises of combination of the two network structures, i.e., upper side and lower side. Where, upper side encloses wavelet neural network and lower side encloses fuzzy reasoning process.

The whole network works in a layered fashion, as follows;

**Layer 1:** This is the first layer of fuzzy reasoning as well as the wavelet network. This layer accepts input values. Its nodes transmit input values to the next layer.

**Layer 2:** In this layer fuzzification process is performed and neurons represent fuzzy sets used in the antecedent part of the linguistic fuzzy rules. The outputs of this layer are the values of the membership functions '*ηj*(*xj*)'.

**Layer 3:** In this layer each node represents a fuzzy rule. In order to compute the firing strength of each rule, and min operation is used to estimate the output value of the layer. i.e.,

$$\mu\_j(\mathbf{x}) = \prod\_i \eta\_j(\mathbf{x}\_i) \tag{3}$$

where, ∏ *i* is the min operation and *μj*(*x*) are the input values for the next layer (consequent layer).

**Layer 4:** In this layer, wavelet functions are represented. The output of this layer is given by;

$$y\_l = w\_l \psi\_l(q) \tag{4}$$

Where, *ψ<sup>l</sup>* = *f*(*ail*, *qil*) is a functional such that

$$q\_{il} = f(x\_{i\prime}b\_{il\prime}a\_{il})$$

The performance index can be expressed as;

*<sup>J</sup>* <sup>=</sup> <sup>1</sup> 2 *O* ∑ *i*=0 *e* 2

> <sup>=</sup> <sup>1</sup> 2 *O* ∑ *i*=0

*wl*(*t* + 1) = *wl*(*t*) − *γ*

*ail*(*t* + 1) = *ail*(*t*) − *γ*

*bil*(*t* + 1) = *bil*(*t*) − *γ*

*gij*(*t* + 1) = *gij*(*t*) − *γ*

*σij*(*t* + 1) = *σij*(*t*) − *γ*

*∂J ∂wl*

*∂J ∂ail*

*∂J ∂bil*

*∂J ∂gij*

*∂J ∂σij* <sup>=</sup> *<sup>∂</sup><sup>J</sup> ∂u ∂u ∂yl*

<sup>=</sup> *<sup>∂</sup><sup>J</sup> ∂u ∂u ∂yl*

<sup>=</sup> *<sup>∂</sup><sup>J</sup> ∂u ∂u ∂yl*

= ∑ *j*

= ∑ *j*

the controller as well so that it could efficiently deal with a nonlinear system.

*∂J ∂u ∂u ∂μ<sup>j</sup>*

*∂J ∂u ∂u ∂μ<sup>j</sup>*

Equations (12) to (16) shows the contribution of update parameters for change in error. The following sections give a brief detail of different configurations of AFWNN, applied to full car model. Since, the full car active suspension control is a nonlinear problem, the idea is to check different combinations of wavelets and membership functions to increase the nonlinearity of

(*ri* − *ui*)

Where, '*ri*' and '*ui*' are the desired and current output values of the system, respectively. '*O*' shows the number of the output values of the system, which is one in our case. The update parameters '*wl*', '*ail*', '*bil*' of the consequent part of network and '*gil*' and '*σil*' (*i* = 1, 2, ..., *m*, *j* = 1, 2, ..., *n*) of the antecedent part of the network can be formulated as follows;

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 153

*∂J ∂wl*

*∂J ∂ail*

*∂J ∂bil*

*∂J ∂gij*

*∂J ∂σij*

Where, '*γ*' and '*λ*' represent the learning rate and momentum, respectively. '*m*' and '*n*' shows the input values and rules number of the network such that *i* = 1, 2, ..., *m* and *j* = 1, 2, ..., *n*. By using chain rule the partial derivatives shown in the above equations can be expanded as;

> *∂yl ∂wl*

*∂yl ∂ψ<sup>l</sup>*

*∂yl ∂ψ<sup>l</sup>*

*∂ψ<sup>l</sup> ∂qil*

*∂ψ<sup>l</sup> ∂ql*

*∂μ<sup>j</sup> ∂gij*

*∂μ<sup>j</sup> ∂σij* *∂qil ∂ail*

*∂ql ∂bl*

<sup>2</sup> (6)

+ *λ*(*wl*(*t*) − *wl*(*t* − 1)) (7)

+ *λ*(*ail*(*t*) − *ail*(*t* − 1)) (8)

+ *λ*(*bil*(*t*) − *bil*(*t* − 1)) (9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

#### Fig. 1. Structure of AFWNN

Here, '*bil*' and '*ail*' represent the parameters for the '*ith*' input and '*lth*' output of the wavelet function. Where, *i* = 1, 2, ..., *n* and *l* = 1, 2, ..., *n*.

**Layer 5:** This layer estimates the weighted consequent value of a given rule.

**Layer 6, 7:** In these layers, the defuzzification process is made to calculate the output of the entire network, i.e., it computes the overall output of system. Therefore, the output for the fuzzy wavelet neural network can be expressed as;

$$\mu = \frac{\sum\_{l=1}^{n} \mu\_l(\mathbf{x}) y\_l}{\sum\_{l=1}^{n} \mu\_l(\mathbf{x})} \tag{5}$$

Where, '*u*' is the output for the entire network. The training of the network starts after estimating the output value of the AFWNN.

The AFWNN learning is to minimize a given function or input and output values by adjusting network parameters. Adapted parameters are mean '*gij*' and variance '*σij*' of membership functions in antecedent part, translation '*bij*' and dilation '*aij*' parameters of wavelet functions and weights '*wij*' are the parameters in the consequent part of the rules.

The AFWNN learning is done by minimizing the performance index. In this study, the gradient descent technique has been used to speed up the convergence and minimize the cost function.

The performance index can be expressed as;

6 Will-be-set-by-IN-TECH

Here, '*bil*' and '*ail*' represent the parameters for the '*ith*' input and '*lth*' output of the wavelet

**Layer 6, 7:** In these layers, the defuzzification process is made to calculate the output of the entire network, i.e., it computes the overall output of system. Therefore, the output for the

*μl*(*x*)*yl*

*μl*(*x*)

(5)

*n* ∑ *l*=1

> *n* ∑ *l*=1

Where, '*u*' is the output for the entire network. The training of the network starts after

The AFWNN learning is to minimize a given function or input and output values by adjusting network parameters. Adapted parameters are mean '*gij*' and variance '*σij*' of membership functions in antecedent part, translation '*bij*' and dilation '*aij*' parameters of wavelet functions

The AFWNN learning is done by minimizing the performance index. In this study, the gradient descent technique has been used to speed up the convergence and minimize the

**Layer 5:** This layer estimates the weighted consequent value of a given rule.

*u* =

and weights '*wij*' are the parameters in the consequent part of the rules.

Fig. 1. Structure of AFWNN

function. Where, *i* = 1, 2, ..., *n* and *l* = 1, 2, ..., *n*.

fuzzy wavelet neural network can be expressed as;

estimating the output value of the AFWNN.

cost function.

$$J = \frac{1}{2} \sum\_{i=0}^{O} e^2$$

$$I = \frac{1}{2} \sum\_{i=0}^{O} (r\_i - u\_i)^2 \tag{6}$$

Where, '*ri*' and '*ui*' are the desired and current output values of the system, respectively. '*O*' shows the number of the output values of the system, which is one in our case. The update parameters '*wl*', '*ail*', '*bil*' of the consequent part of network and '*gil*' and '*σil*' (*i* = 1, 2, ..., *m*, *j* = 1, 2, ..., *n*) of the antecedent part of the network can be formulated as follows;

$$w\_l(t+1) = w\_l(t) - \gamma \frac{\partial f}{\partial w\_l} + \lambda (w\_l(t) - w\_l(t-1)) \tag{7}$$

$$a\_{il}(t+1) = a\_{il}(t) - \gamma \frac{\partial f}{\partial a\_{il}} + \lambda (a\_{il}(t) - a\_{il}(t-1)) \tag{8}$$

$$b\_{il}(t+1) = b\_{il}(t) - \gamma \frac{\partial f}{\partial b\_{il}} + \lambda (b\_{il}(t) - b\_{il}(t-1)) \tag{9}$$

$$g\_{i\bar{j}}(t+1) = g\_{i\bar{j}}(t) - \gamma \frac{\partial f}{\partial g\_{i\bar{j}}} \tag{10}$$

$$
\sigma\_{\hat{i}\hat{j}}(t+1) = \sigma\_{\hat{i}\hat{j}}(t) - \gamma \frac{\partial f}{\partial \sigma\_{\hat{i}\hat{j}}} \tag{11}
$$

Where, '*γ*' and '*λ*' represent the learning rate and momentum, respectively. '*m*' and '*n*' shows the input values and rules number of the network such that *i* = 1, 2, ..., *m* and *j* = 1, 2, ..., *n*.

By using chain rule the partial derivatives shown in the above equations can be expanded as;

$$\frac{\partial \bar{g}\_l}{\partial w\_l} = \frac{\partial \bar{u}}{\partial u} \frac{\partial \bar{y}\_l}{\partial y\_l} \frac{\partial \bar{y}\_l}{\partial w\_l} \tag{12}$$

$$\frac{\partial a\_{\rm il}}{\partial \lambda} = \frac{\partial u}{\partial l} \frac{\partial y\_{\rm l}}{\partial u} \frac{\partial \psi\_{\rm l}}{\partial \psi\_{\rm l}} \frac{\partial q\_{\rm il}}{\partial q\_{\rm il}} \frac{\partial q\_{\rm il}}{\partial q\_{\rm il}} \tag{13}$$

$$\frac{\partial \mathfrak{g}\_{il}}{\partial l} = \frac{\partial \mathfrak{u}}{\partial u} \frac{\partial \mathfrak{y}\_{l}}{\partial \mathfrak{y}\_{l}} \frac{\partial \mathfrak{y}\_{l}}{\partial \mathfrak{y}\_{l}} \frac{\partial \mathfrak{y}\_{l}}{\partial q\_{l}} \frac{\partial q\_{l}}{\partial b\_{l}} \tag{14}$$

$$\frac{\partial f}{\partial \mathbf{g}\_{ij}} = \sum\_{\mathbf{j}} \frac{\partial f}{\partial \boldsymbol{\mu}} \frac{\partial \boldsymbol{\mu}}{\partial \boldsymbol{\mu}\_{\mathbf{j}}} \frac{\partial \boldsymbol{\mu}\_{\mathbf{j}}}{\partial \mathbf{g}\_{ij}} \tag{15}$$

$$\frac{\partial f}{\partial \sigma\_{ij}} = \sum\_{\mathbf{j}} \frac{\partial f}{\partial \mu} \frac{\partial \mu}{\partial \mu\_{\mathbf{j}}} \frac{\partial \mu\_{\mathbf{j}}}{\partial \sigma\_{ij}} \tag{16}$$

Equations (12) to (16) shows the contribution of update parameters for change in error. The following sections give a brief detail of different configurations of AFWNN, applied to full car model. Since, the full car active suspension control is a nonlinear problem, the idea is to check different combinations of wavelets and membership functions to increase the nonlinearity of the controller as well so that it could efficiently deal with a nonlinear system.

*∂J ∂gij*

*∂J ∂σij*

update equations for AFWNN-1 as follows;

*ail*(*t* + 1) = *ail*(*t*) − *γδ<sup>l</sup>*

*bil*(*t* + 1) = *bil*(*t*) − *γδ<sup>l</sup>*

*gij*(*<sup>t</sup>* + <sup>1</sup>) = *gij*(*t*) − ∑

*<sup>σ</sup>ij*(*<sup>t</sup>* + <sup>1</sup>) = *<sup>σ</sup>ij*(*t*) − ∑

process.

where,

= ∑ *j*

= ∑ *j*

*wl*(*t* + 1) = *wl*(*t*) − *γ*(*u*(*t*) − *r*(*t*))*μl*(*x*).*ψl*(*q*)

<sup>⇒</sup> *ail*(*<sup>t</sup>* <sup>+</sup> <sup>1</sup>) = *ail*(*t*) <sup>−</sup> *<sup>γ</sup>*(*u*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*))*μl*(*x*).*wl*(*q*)

+ *λ*(*bil*(*t*) − *bil*(*t* − 1))

<sup>⇒</sup> *bil*(*<sup>t</sup>* <sup>+</sup> <sup>1</sup>) = *bil*(*t*) <sup>−</sup> *<sup>γ</sup>*(*u*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*))*μl*(*x*).*wl*(*q*)

*j*

*j*

 |*ail*|

(3.5*q*<sup>2</sup>

*il* <sup>−</sup> *<sup>q</sup>*<sup>4</sup>

*n* ∑ *l*=1

*n* ∑ *l*=1

(*u*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*)) *yj* <sup>−</sup> *<sup>u</sup>*

(*u*(*t*) − *r*(*t*)).

 *a*3 *il*

*μl*(*x*)

<sup>−</sup>1/2(−3*qil* <sup>+</sup> *<sup>q</sup>*<sup>3</sup>

*μl*(*x*)

∑ *j μj*

*yj* − *u* ∑ *j μj*(*x*)

The gradient descent method shows convergence on the basis of the learning rate and the momentum value. The values of the learning rate and momentum are usually taken in interval [0,1]. If the value of the learning rate is high, it makes the system unstable and if its value is small the convergence process is slow. The momentum term '*λ*� speeds up the learning

(*u*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*)) *yj* <sup>−</sup> *<sup>u</sup>*

(*u*(*t*) − *r*(*t*)).

*δ<sup>l</sup>* = (*u*(*t*) − *r*(*t*))*μl*(*x*).*wl*

∑ *j μj*

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 155

Putting these values in respective equations from equation (7) to equation (11), gives the final

*il* <sup>−</sup> 0.5)*e*−*q*<sup>2</sup>

*yj* − *u* ∑ *j μj*(*x*)

*μj*(*xi*)

.*uj*(*xi*)

 *n* ∑ *l*=1

 *n* ∑ *l*=1

*il*/2

(3.5*q*<sup>2</sup>

+ *λ*(*ail*(*t*) − *ail*(*t* − 1)) (25)

*il*)*e* <sup>−</sup>*q*<sup>2</sup>

> |*ail*|

+ *λ*(*bil*(*t*) − *bil*(*t* − 1)) (26)

2(*xi* − *gij*) *σ*2 *ij*

> <sup>2</sup>(*xi* <sup>−</sup> *gij*)<sup>2</sup> *σ*3 *ij*

*μj*(*xi*)

.*uj*(*xi*)

*il* <sup>−</sup> *<sup>q</sup>*<sup>4</sup>

*il*/2<sup>−</sup><sup>1</sup> *ail*

 *a*3 *il*

<sup>−</sup>1/2(−3*qil* <sup>+</sup> *<sup>q</sup>*<sup>3</sup>

2(*xi* − *gij*) *σ*2 *ij*

*μl*(*x*)

<sup>2</sup>(*xi* <sup>−</sup> *gij*)<sup>2</sup> *σ*3 *ij*

+ *λ*(*ail*(*t*) − *ail*(*t* − 1))

*il* <sup>−</sup> 0.5)*e*−*q*<sup>2</sup>

*μl*(*x*) + *λ*(*wl*(*t*) − *wl*(*t* − 1)) (24)

*il*/2

*il*)*e* <sup>−</sup>*q*<sup>2</sup>

*il*/2<sup>−</sup><sup>1</sup> *ail*

(27)

(28)

(22)

(23)

#### **2.1 AFWNN-1: Structure and parameters update rules for learning**

AFWNN-1 structure uses Gaussian membership function in the antecedent part and Mexican hat wavelet in the consequent part. The gaussian membership function is given by;

$$\eta\_{\dot{j}}(\mathbf{x}\_{\dot{l}}) = e^{-(\mathbf{x}\_{\dot{l}} - \mathbf{g}\_{\dot{l}\dot{l}})^2 / \sigma\_{\dot{l}\dot{l}}^2} \quad \mathbf{i} = 1, 2, \dots, m, \quad \mathbf{j} = 1, 2, \dots, n \tag{17}$$

Where, '*ηj*(*xj*)' shows the membership function, '*gij*' and '*σij*' are the mean and variance of membership function of the *jith* term of *ith* input variable. '*m*' and '*n*' are the number of input signals and number of nodes in second layer, respectively.

The Mexican hat wavelet function is given by;

$$\psi(q\_i) = \sum\_{i=1}^{m} |a\_i|^{-1/2} (1 - q\_i^2) e^{-q\_i^2/2}$$

where,

$$q\_{\bar{j}} = \frac{\mathbf{x} - b\_{\bar{j}}}{a\_{\bar{j}}} \tag{18}$$

Where, Ψ*j*(*x*) is the family of wavelets, *x* = *x*1, *x*2, ..., *xm* shows the inputs values, *aj* = *a*1*j*, *a*2*j*, ..., *amj* and *bj* = *b*1*j*, *b*2*j*, ..., *bmj* represent the dilation and translation parameters of the mother wavelet Ψ(*x*), respectively. Figure 2(a) shows Mexican wavelet function.

#### Fig. 2. Wavelet functions

Referring to equations (12) to (16) and simplifying gives the following results;

$$\frac{\partial f}{\partial w\_l} = (\mu(t) - r(t))\mu\_l(\mathbf{x}).\psi(q\_l) \bigg/ \sum\_{l=1}^{n} \mu\_l(\mathbf{x})\tag{19}$$

$$\frac{\partial J}{\partial a\_{il}} = \delta\_i \frac{(-3.5q\_{il}^2 - q\_{il}^4 - 0.5)e^{-q\_{il}^2/2}}{\sqrt{a\_{il}^3}} \tag{20}$$

$$\frac{\partial J}{\partial b\_{il}} = \delta\_l (3q\_{il} - q\_{il}^3) e^{-q\_{il}^2/2} \left/ \left(\sqrt{a\_{il}^3}\right) \right. \tag{21}$$

$$\frac{\partial f}{\partial \mathcal{g}\_{ij}} = \sum\_{j} (u(t) - r(t)) \frac{y\_j - u}{\sum\_{j} \mu\_j} \mu\_j(\mathbf{x}\_i) \frac{2(\mathbf{x}\_i - \mathbf{g}\_{ij})}{\sigma\_{ij}^2} \tag{22}$$

$$\frac{\partial f}{\partial \sigma\_{ij}} = \sum\_{j} (u(t) - r(t)) \frac{y\_j - u}{\sum\_{j} \mu\_j(\mathbf{x})} . u\_j(\mathbf{x}\_i) \frac{2(\mathbf{x}\_i - g\_{ij})^2}{\sigma\_{ij}^3} \tag{23}$$

where,

8 Will-be-set-by-IN-TECH

AFWNN-1 structure uses Gaussian membership function in the antecedent part and Mexican

Where, '*ηj*(*xj*)' shows the membership function, '*gij*' and '*σij*' are the mean and variance of membership function of the *jith* term of *ith* input variable. '*m*' and '*n*' are the number of input

<sup>−</sup>1/2(<sup>1</sup> <sup>−</sup> *<sup>q</sup>*<sup>2</sup>

*qj* <sup>=</sup> *<sup>x</sup>* <sup>−</sup> *bj aj*

Where, Ψ*j*(*x*) is the family of wavelets, *x* = *x*1, *x*2, ..., *xm* shows the inputs values, *aj* = *a*1*j*, *a*2*j*, ..., *amj* and *bj* = *b*1*j*, *b*2*j*, ..., *bmj* represent the dilation and translation parameters of the

(a) Mexican hat (b) Morlet

= (*u*(*t*) − *r*(*t*))*μl*(*x*).*ψ*(*ql*)

*il* <sup>−</sup> *<sup>q</sup>*<sup>4</sup>

*il*)*e* <sup>−</sup>*q*<sup>2</sup> *il*/2

 *a*3 *il*  *n* ∑ *l*=1

*il*/2

*il* <sup>−</sup> 0.5)*e*−*q*<sup>2</sup>

 *a*3 *il*  *μl*(*x*) (19)

Referring to equations (12) to (16) and simplifying gives the following results;

(−3.5*q*<sup>2</sup>

<sup>=</sup> *<sup>δ</sup>l*(3*qil* <sup>−</sup> *<sup>q</sup>*<sup>3</sup>

*∂J ∂wl*

*∂J ∂ail*

*∂J ∂bil* = *δ<sup>i</sup>*

*i* )*e* <sup>−</sup>*q*<sup>2</sup> *<sup>i</sup>* /2

*ij i* = 1, 2, ..., *m*, *j* = 1, 2, ..., *n* (17)

(18)

(20)

(21)

hat wavelet in the consequent part. The gaussian membership function is given by;

**2.1 AFWNN-1: Structure and parameters update rules for learning**

<sup>−</sup>(*xi*−*gij*)2/*σ*<sup>2</sup>

*ηj*(*xi*) = *e*

The Mexican hat wavelet function is given by;

where,

Fig. 2. Wavelet functions

signals and number of nodes in second layer, respectively.

*ψ*(*qi*) =

*m* ∑ *i*=1 |*ai*|

mother wavelet Ψ(*x*), respectively. Figure 2(a) shows Mexican wavelet function.

$$\delta\_l = (\mu(t) - r(t))\mu\_l(\boldsymbol{x}).w\_l \bigg/ \sum\_{l=1}^n \mu\_l(\boldsymbol{x}),$$

Putting these values in respective equations from equation (7) to equation (11), gives the final update equations for AFWNN-1 as follows;

$$w\_l(t+1) = w\_l(t) - \gamma(u(t) - r(t))\mu\_l(\mathbf{x}).\psi\_l(q) \Big/ \sum\_{l=1}^n \mu\_l(\mathbf{x}) + \lambda(w\_l(t) - w\_l(t-1)) \tag{24}$$

$$a\_{il}(t+1) = a\_{il}(t) - \gamma \delta\_l \frac{(3.5q\_{il}^2 - q\_{il}^4 - 0.5)e^{-q\_{il}^2/2}}{\sqrt{a\_{il}^3}} + \lambda (a\_{il}(t) - a\_{il}(t-1))$$

$$\begin{split} \Rightarrow a\_{il}(t+1) &= a\_{il}(t) - \frac{\gamma(u(t) - r(t))\mu\_l(\mathbf{x}).w\_l(\mathbf{q})}{\sum\limits\_{l=1}^{n} \mu\_l(\mathbf{x})} \frac{(3.5q\_{il}^2 - q\_{il}^4 - 0.5)e^{-q\_{il}^2/2}}{\sqrt{a\_{il}^3}} \\ &+ \lambda (a\_{il}(t) - a\_{il}(t-1)) \end{split} \tag{25}$$

$$\begin{aligned} b\_{il}(t+1) &= b\_{il}(t) - \gamma \delta\_l \left[ |a\_{il}|^{-1/2} (-3q\_{il} + q\_{il}^3) e^{-q\_{il}^2/2} \left( \frac{-1}{a\_{il}} \right) \right] \\ &+ \lambda (b\_{il}(t) - b\_{il}(t-1)) \\ \Rightarrow b\_{il}(t+1) &= b\_{il}(t) - \frac{\gamma (u(t) - r(t)) \mu\_l(x) \omega\_l(q)}{\sum\limits\_{l=1}^{n} \mu\_l(x)} \left[ |a\_{il}|^{-1/2} (-3q\_{il} + q\_{il}^3) e^{-q\_{il}^2/2} \left( \frac{-1}{a\_{il}} \right) \right] \\ &+ \lambda (b\_{il}(t) - b\_{il}(t-1)) \end{aligned} \tag{26}$$

$$g\_{\bar{i}\bar{j}}(t+1) = g\_{\bar{i}\bar{j}}(t) - \sum\_{\bar{j}} (u(t) - r(t)) \frac{y\_{\bar{j}} - u}{\sum\_{\bar{j}} \mu\_{\bar{j}}} \mu\_{\bar{j}}(\mathbf{x}\_{\bar{i}}) \frac{2(\mathbf{x}\_{\bar{i}} - \mathbf{g}\_{\bar{i}\bar{j}})}{\sigma\_{\bar{i}\bar{j}}^2} \tag{27}$$

$$
\sigma\_{\vec{ij}}(t+1) = \sigma\_{\vec{ij}}(t) - \sum\_{j} (u(t) - r(t)) \frac{y\_j - u}{\sum\_{j} \mu\_j(\mathbf{x})} \mu\_j(\mathbf{x}\_i) \frac{\mathbf{2}(\mathbf{x}\_i - g\_{\vec{ij}})^2}{\sigma\_{\vec{ij}}^3} \tag{28}
$$

The gradient descent method shows convergence on the basis of the learning rate and the momentum value. The values of the learning rate and momentum are usually taken in interval [0,1]. If the value of the learning rate is high, it makes the system unstable and if its value is small the convergence process is slow. The momentum term '*λ*� speeds up the learning process.

<sup>⇒</sup> *bil*(*<sup>t</sup>* <sup>+</sup> <sup>1</sup>) = *bil*(*t*) <sup>−</sup> *<sup>γ</sup>*(*u*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*))*μl*(*x*).*wl*(*q*)

*j*

*j*

function has been shown in Figure 2(b) and is given by;

**2.3 AFWNN-3: Structure and parameters update rules for learning**

*yl* = *wlψl*(*q*), *ψl*(*q*) =

*m* ∑ *i*=1

By using equations (12) to (16), the partial derivatives can be solved as follows;

= (*u*(*t*) − *r*(*t*))*μl*(*x*).*ψ*(*ql*)

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*<sup>u</sup>*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*)

(*u*(*t*) − *r*(*t*)).

⇒ *yl* = *wl*

*∂J ∂wl*

*∂J ∂ail*

*∂J ∂bil*

*∂J ∂gij*

*∂J ∂σij* = *δ<sup>l</sup>*

= *δ<sup>l</sup>*

= ∑ *j*

= ∑ *j*

*gij*(*<sup>t</sup>* + <sup>1</sup>) = *gij*(*t*) − *<sup>γ</sup>*∑

*<sup>σ</sup>ij*(*<sup>t</sup>* + <sup>1</sup>) = *<sup>σ</sup>ij*(*t*) − *<sup>γ</sup>*∑

wavelet network is given by;

respectively.

*n* ∑ *l*=1

*<sup>u</sup>*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*)

*<sup>u</sup>*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*).

*μl*(*x*)

 . *yj* − *u* ∑ *j μj*

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 157

Hence, these are the required equations for the update parameters, '*wl*', '*ail*', '*bil*', '*gil*' and '*σil*'

In AFWNN-3 the consequent part uses Morlet wavelet function whereas the antecedent part uses the same Gaussian membership function as that of AFWNN-1. The Morlet wavelet

The Gaussian membership function is given by equation (17); The output value '*y*' for the '*lth*'

*m* ∑ *i*=1

*m* ∑ *i*=1

*ail e* − 1 2 *xi*−*bil ail* <sup>2</sup>

> *n* ∑ *l*=1

*il* <sup>+</sup> <sup>5</sup>*qil* sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*il*)*qil* <sup>+</sup> 5 sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*μj*(*xi*)

.*uj*(*xi*)

*ail*

*ail*

− 1 <sup>2</sup> (*q*<sup>2</sup>

cos(5*qil*)*e*

cos(5*qil*)*e*

− 1 <sup>2</sup> (*q*<sup>2</sup> *il*)

− 1 <sup>2</sup> (*q*<sup>2</sup> *il*)

Ψ*j*(*x*) = *cos*(5*qj*)*e*

⇒ *yl* = *wl*

cos 5 *xi* <sup>−</sup> *bil*

<sup>2</sup> (*q*<sup>2</sup> *il*)*q*<sup>2</sup>

<sup>2</sup> (*q*<sup>2</sup>

 *yj* − *u* ∑ *j μj*

> *yj* − *u* ∑ *j μj*(*x*)

 *yj* − *u* ∑ *j μj*

 |*ail*|

+ *λ*(*bil*(*t*) − *bil*(*t* − 1)) (37)

*μj ηj*(*xi*)

. *<sup>μ</sup><sup>j</sup> ηj*(*xi*)

<sup>−</sup>1/2(−3*qil* <sup>+</sup> *<sup>q</sup>*<sup>3</sup>

1 − *η<sup>j</sup> σij*

.2 *sign*(*xi* <sup>−</sup> *gij*) *<sup>σ</sup>ij*

*il*)*e* <sup>−</sup>*q*<sup>2</sup>

*<sup>j</sup>* ) (40)

*μl*(*x*) (42)

<sup>2</sup> (*q*<sup>2</sup> *il*)

<sup>2</sup> (*q*<sup>2</sup> *il*)

2(*xi* − *gij*) *σ*2 *ij*

> <sup>2</sup>(*xi* <sup>−</sup> *gij*)<sup>2</sup> *σ*3 *ij*

*il*/2<sup>−</sup><sup>1</sup> *ail*

(38)

(39)

(41)

(43)

(44)

(45)

(46)

#### **2.2 AFWNN-2: Structure and parameters update rules for learning**

In AFWNN-2, linear function or constant in the consequent part of the linguistic rules in TSK fuzzy system are replaced with Mexican hat wavelet function. The Mexican Hat wavelet function is given by equation (18), as for AFWNN-1. To illustrate the linguistic term, the Triangular membership function has been used for this neuro-fuzzy system and is given by,

$$\eta\_{\bar{j}}(\mathbf{x}\_{i}) = 1 - \frac{2 \mid \mathbf{x}\_{i} - \mathbf{g}\_{i\bar{j}} \mid}{\sigma\_{\text{ij}}} \quad i = 1, 2, \ldots, m, \quad j = 1, 2, \ldots, n \tag{29}$$

Where, '*ηj*(*xj*)' shows the membership function, '*gij*' and '*σij*' are the mean and variance of membership function of the '*jith*' term of '*ith*' input variable. In order to calculate the updated values for this network simplifying the Equations (12) to (16) give the following results;

$$\frac{\partial f}{\partial w\_l} = (u(t) - r(t))\mu\_l(\mathbf{x}).\psi(q\_l) \bigg/ \sum\_{l=1}^n \mu\_l(\mathbf{x})\tag{30}$$

$$\frac{\partial f}{\partial a\_{il}} = \delta\_l \frac{(-3.5q\_{il}^2 - q\_{il}^4 - 0.5)e^{-q\_{il}^2/2}}{\sqrt{a\_{il}^3}} \tag{31}$$

$$\frac{\partial f}{\partial b\_{il}} = \delta\_l (3q\_{il} - q\_{il}^3) e^{-q\_{il}^2/2} \left/ \left(\sqrt{a\_{il}^3}\right)\right. \tag{32}$$

$$\frac{\partial f}{\partial \mathcal{g}\_{ij}} = \sum\_{j} \left[ \left( u(t) - r(t) \right) . \frac{y\_j - u}{\sum\_{j} \mu\_j} \frac{\mu\_j}{\eta\_j(\mathbf{x}\_i)} . 2 \frac{\operatorname{sign}(\mathbf{x}\_i - \mathbf{g}\_{ij})}{\sigma\_{ij}} \right] \tag{33}$$

$$\frac{\partial \bar{l}}{\partial \sigma\_{ij}} = \sum\_{\bar{j}} \left[ \left( u(t) - r(t). \right) \frac{y\_{\bar{j}} - u}{\sum\_{\bar{j}} \mu\_{\bar{j}}} \cdot \frac{\mu\_{\bar{j}}}{\eta\_{\bar{j}}(\mathbf{x}\_{i})} \frac{1 - \eta\_{\bar{j}}}{\sigma\_{ij}} \right] \tag{34}$$

By putting these values in Equations (7) to (11) the final update equations are given by;

$$\begin{aligned} w\_{l}(t+1) &= w\_{l}(t) - \gamma(u(t) - r(t))\mu\_{l}(x)\,\upvarphi\_{l}(q) \bigg/ \sum\_{l=1}^{n} \mu\_{l}(x) \\ &+ \lambda (w\_{l}(t) - w\_{l}(t-1)) \\ a\_{il}(t+1) &= a\_{il}(t) - \gamma \delta\_{l} \frac{(-3.5q\_{ll}^{2} - q\_{ll}^{4} - 0.5)e^{-q\_{ll}^{2}/2}}{\sqrt{a\_{il}^{3}}} \\ &+ \lambda (a\_{il}(t) - a\_{il}(t-1)) \\ \Rightarrow a\_{il}(t+1) &= a\_{il}(t) - \frac{\gamma(u(t) - r(t))\mu\_{l}(x).w\_{l}(q)}{\sum\_{l=1}^{n} \mu\_{l}(x)} \frac{(-3.5q\_{ll}^{2} - q\_{ll}^{4} - 0.5)e^{-q\_{ll}^{2}/2}}{\sqrt{a\_{il}^{3}}} \\ &+ \lambda (a\_{il}(t) - a\_{il}(t-1)) \\ b\_{ll}(t+1) &= b\_{ll}(t) - \gamma \delta\_{l} \Big[|a\_{il}|^{-1/2}(-3q\_{ll} + q\_{ll}^{3})e^{-q\_{ll}^{2}/2} \Big(\frac{-1}{a\_{il}}\Big{)}\Big{]} \\ &+ \lambda (b\_{il}(t) - b\_{il}(t-1)) \end{aligned} \tag{36}$$

10 Will-be-set-by-IN-TECH

In AFWNN-2, linear function or constant in the consequent part of the linguistic rules in TSK fuzzy system are replaced with Mexican hat wavelet function. The Mexican Hat wavelet function is given by equation (18), as for AFWNN-1. To illustrate the linguistic term, the Triangular membership function has been used for this neuro-fuzzy system and is given by,

Where, '*ηj*(*xj*)' shows the membership function, '*gij*' and '*σij*' are the mean and variance of membership function of the '*jith*' term of '*ith*' input variable. In order to calculate the updated values for this network simplifying the Equations (12) to (16) give the following results;

*il* <sup>−</sup> 0.5)*e*−*q*<sup>2</sup>

 . *yj* − *u* ∑ *j μj*

By putting these values in Equations (7) to (11) the final update equations are given by;

*il* <sup>−</sup> *<sup>q</sup>*<sup>4</sup>

 *a*3 *il*

*μl*(*x*)

<sup>−</sup>1/2(−3*qil* <sup>+</sup> *<sup>q</sup>*<sup>3</sup>

 *a*3 *il* 

 *yj* − *u* ∑ *j μj*

 *n* ∑ *l*=1

*il*/2

*μj ηj*(*xi*)

. *<sup>μ</sup><sup>j</sup> ηj*(*xi*)

.2 *sign*(*xi* <sup>−</sup> *gij*) *σij*

*μl*(*x*)

1 − *η<sup>j</sup> σij*

 *n* ∑ *l*=1

*il*/2

(−3.5*q*<sup>2</sup>

*il* <sup>−</sup> *<sup>q</sup>*<sup>4</sup>

 *a*3 *il*

*il* <sup>−</sup> 0.5)*e*−*q*<sup>2</sup>

*il*/2

+ *λ*(*wl*(*t*) − *wl*(*t* − 1)) (35)

+ *λ*(*ail*(*t*) − *ail*(*t* − 1)) (36)

*il*)*e* <sup>−</sup>*q*<sup>2</sup> *il*/2 <sup>−</sup><sup>1</sup> *ail*

*il* <sup>−</sup> 0.5)*e*−*q*<sup>2</sup>

*i* = 1, 2, ..., *m*, *j* = 1, 2, ..., *n* (29)

*μl*(*x*) (30)

(31)

(32)

(33)

(34)

**2.2 AFWNN-2: Structure and parameters update rules for learning**

*<sup>η</sup>j*(*xi*) = <sup>1</sup> <sup>−</sup> <sup>2</sup> <sup>|</sup> *xi* <sup>−</sup> *gij* <sup>|</sup>

(−3.5*q*<sup>2</sup>

<sup>=</sup> *<sup>δ</sup>l*(3*qil* <sup>−</sup> *<sup>q</sup>*<sup>3</sup>

*∂J ∂wl*

*∂J ∂ail*

*∂J ∂bil*

*∂J ∂gij*

*∂J ∂σij* = *δ<sup>l</sup>*

= ∑ *j*

= ∑ *j*

*ail*(*t* + 1) = *ail*(*t*) − *γδ<sup>l</sup>*

*bil*(*t* + 1) = *bil*(*t*) − *γδ<sup>l</sup>*

*σij*

= (*u*(*t*) − *r*(*t*))*μl*(*x*).*ψ*(*ql*)

*il* <sup>−</sup> *<sup>q</sup>*<sup>4</sup>

*il*)*e* <sup>−</sup>*q*<sup>2</sup> *il*/2

*u*(*t*) − *r*(*t*)

*u*(*t*) − *r*(*t*).

(−3.5*q*<sup>2</sup>

*n* ∑ *l*=1

*wl*(*t* + 1) = *wl*(*t*) − *γ*(*u*(*t*) − *r*(*t*))*μl*(*x*).*ψl*(*q*)

+ *λ*(*ail*(*t*) − *ail*(*t* − 1))

<sup>⇒</sup> *ail*(*<sup>t</sup>* <sup>+</sup> <sup>1</sup>) = *ail*(*t*) <sup>−</sup> *<sup>γ</sup>*(*u*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*))*μl*(*x*).*wl*(*q*)

+ *λ*(*bil*(*t*) − *bil*(*t* − 1))

 |*ail*|

 *a*3 *il*

$$0 \Rightarrow b\_{il}(t+1) = b\_{il}(t) - \frac{\gamma(u(t) - r(t))\mu\_l(\mathbf{x}).w\_l(\mathbf{q})}{\sum\_{l=1}^{n} \mu\_l(\mathbf{x})} \left[|a\_{il}|^{-1/2}(-3q\_{il} + q\_{il}^3)e^{-q\_{il}^2/2} \left(\frac{-1}{a\_{il}}\right)\right]$$

$$\begin{array}{ccccccccc}\hline\\+&\lambda\left(b\_{il}(t)-b\_{il}(t-1)\right)&\\&\ddots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\gamma\end{array}\tag{37}$$

$$g\_{ij}(t+1) = g\_{ij}(t) - \gamma \sum\_{j} \left[ \left( u(t) - r(t) \right) . \frac{y\_j - u}{\sum\_{j} \mu\_j} \frac{\mu\_j}{\eta\_j(\mathbf{x}\_i)} . 2 \frac{\operatorname{sign}(\mathbf{x}\_i - \mathbf{g}\_{ij})}{\sigma\_{ij}} \right] \tag{38}$$

$$
\sigma\_{\vec{l}\vec{j}}(t+1) = \sigma\_{\vec{l}\vec{j}}(t) - \gamma \sum\_{\vec{j}} \left[ \left( u(t) - r(t). \right) \frac{y\_{\vec{j}} - u}{\sum\_{\vec{j}} \mu\_{\vec{j}}} \cdot \frac{\mu\_{\vec{j}}}{\eta\_{\vec{j}}(\mathbf{x}\_{\vec{l}})} \frac{1 - \eta\_{\vec{j}}}{\sigma\_{\vec{l}\vec{j}}} \right] \tag{39}
$$

Hence, these are the required equations for the update parameters, '*wl*', '*ail*', '*bil*', '*gil*' and '*σil*' respectively.

#### **2.3 AFWNN-3: Structure and parameters update rules for learning**

In AFWNN-3 the consequent part uses Morlet wavelet function whereas the antecedent part uses the same Gaussian membership function as that of AFWNN-1. The Morlet wavelet function has been shown in Figure 2(b) and is given by;

$$\Psi\_{\dot{\!\!\!/}}(\mathbf{x}) = \cos(5q\_{\!\!/})e^{-\frac{1}{2}\left(q\_{\!\!/}^{2}\right)} \tag{40}$$

The Gaussian membership function is given by equation (17); The output value '*y*' for the '*lth*' wavelet network is given by;

$$\begin{aligned} y\_l &= w\_l \psi\_l(q), \quad \psi\_l(q) = \sum\_{i=1}^m \cos(5q\_{il}) e^{-\frac{1}{2}(q\_{il}^2)} \\ &\Rightarrow y\_l = w\_l \sum\_{i=1}^m \cos(5q\_{il}) e^{-\frac{1}{2}(q\_{il}^2)} \end{aligned}$$

$$\Rightarrow y\_l = w\_l \sum\_{i=1}^m \cos 5 \left(\frac{x\_i - b\_{il}}{a\_{il}}\right) e^{-\frac{1}{2}\left(\frac{x\_i - b\_{il}}{a\_{il}}\right)^2} \tag{41}$$

By using equations (12) to (16), the partial derivatives can be solved as follows;

$$\frac{\partial f}{\partial w\_l} = (\mu(t) - r(t))\mu\_l(\mathbf{x}).\psi(q\_l) \bigg/ \sum\_{l=1}^n \mu\_l(\mathbf{x})\tag{42}$$

$$\frac{\partial J}{\partial a\_{il}} = \delta\_l \left( \frac{\cos(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}q\_{il}^2 + 5q\_{il}\sin(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}}{a\_{il}} \right) \tag{43}$$

$$\frac{\partial J}{\partial b\_{il}} = \delta\_l \left( \frac{\cos(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}q\_{il} + \dots \sin(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}}{a\_{il}} \right) \tag{44}$$

$$\frac{\partial J}{\partial g\_{ij}} = \sum\_{j} \left[ \left( u(t) - r(t) \right) \frac{y\_j - u}{\sum\_{j} \mu\_j} \mu\_j(\mathbf{x}\_i) \frac{2(\mathbf{x}\_i - g\_{ij})}{\sigma\_{ij}^2} \right] \tag{45}$$

$$\frac{\partial f}{\partial \sigma\_{ij}} = \sum\_{j} \left[ (u(t) - r(t)) . \frac{y\_j - u}{\sum\_{j} \mu\_j(\mathbf{x})} . u\_j(\mathbf{x}\_l) \frac{2(\mathbf{x}\_i - g\_{ij})^2}{\sigma\_{ij}^3} \right] \tag{46}$$

The derivatives given by equations (12) to (16) can be simplified as follows;

 *n* ∑ *l*=1

*il* <sup>+</sup> <sup>5</sup>*qil* sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*il*)*qil* <sup>+</sup> 5 sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*μj ηj*(*xi*)

. *<sup>μ</sup><sup>j</sup> ηj*(*xi*)

*∂wl* , *<sup>∂</sup><sup>J</sup> <sup>∂</sup>ail* , *<sup>∂</sup><sup>J</sup> <sup>∂</sup>bil* , *<sup>∂</sup><sup>J</sup> ∂gij*

 *n* ∑ *l*=1

+ *λ*(*wl*(*t*) − *wl*(*t* − 1)) (58)

+ *λ*(*ail*(*t*) − *ail*(*t* − 1)) (59)

*il*)*qil* <sup>+</sup> 5 sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

+ *λ*(*bil*(*t*) − *bil*(*t* − 1)) (60)

*μj ηj*(*xi*)

. *<sup>μ</sup><sup>j</sup> ηj*(*xi*)

*il* <sup>+</sup> <sup>5</sup>*qil* sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*ail*

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*μl*(*x*)

*ail*

*ail*

1 − *η<sup>j</sup> σij*

*ail*

*μl*(*x*) (53)

(54)

(55)

(56)

(57)

<sup>2</sup> (*q*<sup>2</sup> *il*)

<sup>2</sup> (*q*<sup>2</sup> *il*)

.2 *sign*(*xi* <sup>−</sup> *gij*) *<sup>σ</sup>ij*

and *<sup>∂</sup><sup>J</sup>*

<sup>2</sup> (*q*<sup>2</sup> *il*)

<sup>2</sup> (*q*<sup>2</sup> *il*)*q*<sup>2</sup>

<sup>2</sup> (*q*<sup>2</sup> *il*)

<sup>2</sup> (*q*<sup>2</sup>

.2 *sign*(*xi* <sup>−</sup> *gij*) *<sup>σ</sup>ij*

1 − *η<sup>j</sup> σij* *∂σij* , respectively.

*il* <sup>+</sup> <sup>5</sup>*qil* sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*il*)*qil* <sup>+</sup> 5 sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*ail*

*ail*

<sup>2</sup> (*q*<sup>2</sup> *il*)

<sup>2</sup> (*q*<sup>2</sup> *il*)

(61)

(62)

= (*u*(*t*) − *r*(*t*))*μl*(*x*).*ψ*(*ql*)

<sup>2</sup> (*q*<sup>2</sup> *il*)*q*<sup>2</sup>

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 159

<sup>2</sup> (*q*<sup>2</sup>

 . *yj* − *u* ∑ *j μj*

 *yj* − *u* ∑ *j μj*

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*<sup>u</sup>*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*)

*<sup>u</sup>*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*)

Using Equations (7) to (11) the updates can be found as follows;

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*n* ∑ *l*=1

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*n* ∑ *l*=1

*<sup>u</sup>*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*)

*<sup>u</sup>*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*).

*μl*(*x*)

*μl*(*x*)

<sup>2</sup> (*q*<sup>2</sup> *il*)*q*<sup>2</sup>

<sup>2</sup> (*q*<sup>2</sup>

 . *yj* − *u* ∑ *j μj*

 *yj* − *u* ∑ *j μj*

*∂J ∂wl*

*∂J ∂ail*

*∂J ∂bil*

*∂J ∂gij*

*∂J ∂σij*

*ail*(*t* + 1) = *ail*(*t*) − *γδ<sup>l</sup>*

*bil*(*t* + 1) = *bil*(*t*) − *γδ<sup>l</sup>*

*gij*(*<sup>t</sup>* + <sup>1</sup>) = *gij*(*t*) − *<sup>γ</sup>*∑

*<sup>σ</sup>ij*(*<sup>t</sup>* + <sup>1</sup>) = *<sup>σ</sup>ij*(*t*) − *<sup>γ</sup>*∑

= *δ<sup>l</sup>*

= *δ<sup>l</sup>*

= ∑ *j*

= ∑ *j*

Equations (53) to (57) give the required values of *<sup>∂</sup><sup>J</sup>*

*wl*(*t* + 1) = *wl*(*t*) − *γ*(*u*(*t*) − *r*(*t*))*μl*(*x*).*ψl*(*q*)

+ *λ*(*ail*(*t*) − *ail*(*t* − 1))

<sup>⇒</sup> *ail*(*<sup>t</sup>* <sup>+</sup> <sup>1</sup>) = *ail*(*t*) <sup>−</sup> *<sup>γ</sup>*(*u*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*))*μl*(*x*).*wl*(*q*)

+ *λ*(*bil*(*t*) − *bil*(*t* − 1))

<sup>⇒</sup> *bil*(*<sup>t</sup>* <sup>+</sup> <sup>1</sup>) = *bil*(*t*) <sup>−</sup> *<sup>γ</sup>*(*u*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*))*μl*(*x*).*wl*(*q*)

*j*

*j*

Equations (42) to (46) give the required values of *<sup>∂</sup><sup>J</sup> ∂wl* , *<sup>∂</sup><sup>J</sup> <sup>∂</sup>ail* , *<sup>∂</sup><sup>J</sup> <sup>∂</sup>bil* , *<sup>∂</sup><sup>J</sup> ∂gij* and *<sup>∂</sup><sup>J</sup> ∂σij* , showing the contribution of each update parameter for error convergence.

The required updates can be calculated using equations (7) to (11) as follows;

$$w\_l(t+1) = w\_l(t) - \gamma(u(t) - r(t))\mu\_l(\mathbf{x})\,\varphi\_l(\mathbf{q}) \left/ \sum\_{l=1}^n \mu\_l(\mathbf{x}) + \lambda(w\_l(t) - w\_l(t-1)) \right. \tag{47}$$

$$a\_{il}(t+1) = a\_{il}(t) - \gamma \delta\_l \left( \frac{\cos(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}q\_{ll}^2 + 5q\_{il}\sin(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}}{a\_{il}} \right)$$

$$\quad + \lambda(a\_{il}(t) - a\_{il}(t-1))$$

$$\Rightarrow a\_{il}(t+1) = a\_{il}(t) - \frac{\gamma(u(t) - r(t))\mu\_l(\mathbf{x})\,w\_l(q)}{\sum\_{l=1}^n \mu\_l(\mathbf{x})} \left( \frac{\cos(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}q\_{ll}^2 + 5q\_{il}\sin(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}}{a\_{il}} \right)$$

$$\quad + \lambda(a\_{il}(t) - a\_{il}(t-1)) \tag{48}$$

$$b\_{il}(t+1) = b\_{il}(t) - \gamma\delta\_l \left( \frac{\cos(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}q\_{ll} + 5\sin(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}}{a\_{il}} \right)$$

$$\quad + \lambda(b\_{il}(t) - b\_{il}(t-1))$$

$$0 \Rightarrow b\_{\rm il}(t+1) = b\_{\rm il}(t) - \frac{\gamma(u(t) - r(t))\mu\_l(\mathbf{x})\,\omega\_l(q)}{\sum\_{l=1}^{n} \mu\_l(\mathbf{x})} \left(\frac{\cos(5q\_{\rm ill})e^{-\frac{1}{2}(q\_{\rm il}^2)}q\_{\rm il} + 5\sin(5q\_{\rm ill})e^{-\frac{1}{2}(q\_{\rm il}^2)}}{a\_{\rm il}}\right) \;^3\mathbf{x}$$

$$+\left.\lambda(b\_{il}(t) - b\_{il}(t-1))\right|\_{\Delta^{\prime}}\tag{49}$$

$$g\_{i\bar{j}}(t+1) = g\_{i\bar{j}}(t) - \gamma \sum\_{\bar{j}} u(t) - r(t) \frac{y\_{\bar{j}} - u}{\sum\_{\bar{j}} \mu\_{\bar{j}}} \mu\_{\bar{j}}(\mathbf{x}\_{i}) \frac{\mathbf{2}(\mathbf{x}\_{i} - \mathbf{g}\_{i\bar{j}})}{\sigma\_{i\bar{j}}^{2}} \tag{50}$$

$$
\sigma\_{\vec{ij}}(t+1) = \sigma\_{\vec{ij}}(t) - \gamma \sum\_{j} u(t) - r(t) \frac{y\_j - u}{\sum\_{j} \mu\_j} \mu\_j(\mathbf{x}\_i) \frac{2(\mathbf{x}\_i - \mathbf{g}\_{\vec{ij}})^2}{\sigma\_{\vec{ij}}^3} \tag{51}
$$

Hence, these are the required equations for the update parameters *wl*, *ail*, *bil*, *gil* and *σil*.

#### **2.4 AFWNN-4: Structure and parameters update rules for learning**

AFWNN-4 uses Morlet wavelet function along with triangular membership function. The triangular membership function is given by Equation (29). Using Morlet wavelet function the output value '*y*' for the '*lth*' wavelet is given by;

$$y\_l = w\_l \psi\_l(q), \quad \psi\_l(q) = \sum\_{i=1}^{m} \cos(5q\_{il}) e^{-\frac{1}{2}(q\_{il}^2)}$$

$$\Rightarrow y\_l = w\_l \sum\_{i=1}^{m} \cos 5 \left(\frac{\mathbf{x}\_i - b\_{il}}{a\_{il}}\right) e^{-\frac{1}{2} \left(\frac{\mathbf{x}\_i - b\_{il}}{a\_{il}}\right)^2} \tag{52}$$

12 Will-be-set-by-IN-TECH

*∂wl* , *<sup>∂</sup><sup>J</sup> <sup>∂</sup>ail* , *<sup>∂</sup><sup>J</sup>*

*il* <sup>+</sup> <sup>5</sup>*qil* sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

2(*xi* − *gij*) *σ*2 *ij*

<sup>2</sup>(*xi* <sup>−</sup> *gij*)<sup>2</sup> *σ*3 *ij*

+ *λ*(*bil*(*t*) − *bil*(*t* − 1)) (49)

*μj*(*xi*)

*μj*(*xi*)

+ *λ*(*ail*(*t*) − *ail*(*t* − 1)) (48)

*il*)*qil* <sup>+</sup> 5 sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*ail*

 *n* ∑ *l*=1

*ail*

*<sup>∂</sup>bil* , *<sup>∂</sup><sup>J</sup> ∂gij*

> <sup>2</sup> (*q*<sup>2</sup> *il*)

<sup>2</sup> (*q*<sup>2</sup> *il*)*q*<sup>2</sup>

<sup>2</sup> (*q*<sup>2</sup> *il*)

<sup>2</sup> (*q*<sup>2</sup>

*ail*

and *<sup>∂</sup><sup>J</sup>*

*μl*(*x*) + *λ*(*wl*(*t*) − *wl*(*t* − 1)) (47)

*il* <sup>+</sup> <sup>5</sup>*qil* sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*il*)*qil* <sup>+</sup> 5 sin(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*ail*

*∂σij* , showing the

<sup>2</sup> (*q*<sup>2</sup> *il*)

<sup>2</sup> (*q*<sup>2</sup> *il*)

(50)

(51)

(52)

Equations (42) to (46) give the required values of *<sup>∂</sup><sup>J</sup>*

*wl*(*t* + 1) = *wl*(*t*) − *γ*(*u*(*t*) − *r*(*t*))*μl*(*x*).*ψl*(*q*)

+ *λ*(*ail*(*t*) − *ail*(*t* − 1))

<sup>⇒</sup> *ail*(*<sup>t</sup>* <sup>+</sup> <sup>1</sup>) = *ail*(*t*) <sup>−</sup> *<sup>γ</sup>*(*u*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*))*μl*(*x*).*wl*(*q*)

+ *λ*(*bil*(*t*) − *bil*(*t* − 1))

<sup>⇒</sup> *bil*(*<sup>t</sup>* <sup>+</sup> <sup>1</sup>) = *bil*(*t*) <sup>−</sup> *<sup>γ</sup>*(*u*(*t*) <sup>−</sup> *<sup>r</sup>*(*t*))*μl*(*x*).*wl*(*q*)

*j*

*j*

output value '*y*' for the '*lth*' wavelet is given by;

⇒ *yl* = *wl*

*ail*(*t* + 1) = *ail*(*t*) − *γδ<sup>l</sup>*

*bil*(*t* + 1) = *bil*(*t*) − *γδ<sup>l</sup>*

*gij*(*<sup>t</sup>* + <sup>1</sup>) = *gij*(*t*) − *<sup>γ</sup>*∑

*<sup>σ</sup>ij*(*<sup>t</sup>* + <sup>1</sup>) = *<sup>σ</sup>ij*(*t*) − *<sup>γ</sup>*∑

contribution of each update parameter for error convergence.

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*n* ∑ *l*=1

cos(5*qil*)*e*<sup>−</sup> <sup>1</sup>

*n* ∑ *l*=1

*u*(*t*) − *r*(*t*)

*u*(*t*) − *r*(*t*)

**2.4 AFWNN-4: Structure and parameters update rules for learning**

*yl* = *wlψl*(*q*), *ψl*(*q*) =

cos 5

*m* ∑ *i*=1

*μl*(*x*)

*yj* − *u* ∑ *j μj*

*yj* − *u* ∑ *j μj*

Hence, these are the required equations for the update parameters *wl*, *ail*, *bil*, *gil* and *σil*.

AFWNN-4 uses Morlet wavelet function along with triangular membership function. The triangular membership function is given by Equation (29). Using Morlet wavelet function the

> *xi* <sup>−</sup> *bil ail*

*m* ∑ *i*=1

> *e* − 1 2 *xi*−*bil ail* 2

cos(5*qil*)*e*

− 1 <sup>2</sup> (*q*<sup>2</sup> *il*)

*μl*(*x*)

<sup>2</sup> (*q*<sup>2</sup>

The required updates can be calculated using equations (7) to (11) as follows;

<sup>2</sup> (*q*<sup>2</sup> *il*)*q*<sup>2</sup> The derivatives given by equations (12) to (16) can be simplified as follows;

$$\frac{\partial f}{\partial w\_l} = (u(t) - r(t))\mu\_l(\mathbf{x}).\psi(q\_l) \bigg/ \sum\_{l=1}^n \mu\_l(\mathbf{x})\tag{53}$$

$$\frac{\partial J}{\partial a\_{il}} = \delta\_l \left( \frac{\cos(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}q\_{il}^2 + 5q\_{il}\sin(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}}{a\_{il}} \right) \tag{54}$$

$$\frac{\partial J}{\partial b\_{il}} = \delta\_l \left( \frac{\cos(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}q\_{il} + \dots \sin(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}}{a\_{il}} \right) \tag{55}$$

$$\frac{\partial f}{\partial \mathbf{g}\_{ij}} = \sum\_{j} \left[ \left( u(t) - r(t) \right) . \frac{y\_j - u}{\sum\_{j} \mu\_j} \frac{\mu\_j}{\eta\_j(\mathbf{x}\_i)} . 2 \frac{\operatorname{sign}(\mathbf{x}\_i - \mathbf{g}\_{ij})}{\sigma\_{ij}} \right] \tag{56}$$

$$\frac{\partial f}{\partial \sigma\_{ij}} = \sum\_{j} \left[ \left( u(t) - r(t) \right) \frac{y\_j - u}{\sum\_{j} \mu\_j} \cdot \frac{\mu\_j}{\eta\_j(\mathbf{x}\_i)} \frac{1 - \eta\_j}{\sigma\_{ij}} \right] \tag{57}$$

Equations (53) to (57) give the required values of *<sup>∂</sup><sup>J</sup> ∂wl* , *<sup>∂</sup><sup>J</sup> <sup>∂</sup>ail* , *<sup>∂</sup><sup>J</sup> <sup>∂</sup>bil* , *<sup>∂</sup><sup>J</sup> ∂gij* and *<sup>∂</sup><sup>J</sup> ∂σij* , respectively. Using Equations (7) to (11) the updates can be found as follows;

$$\begin{split} w\_{l}(t+1) &= w\_{l}(t) - \gamma(u(t) - r(t))\mu\_{l}(\mathbf{x})\,\varphi\_{l}(q) \bigg/ \sum\_{l=1}^{n} \mu\_{l}(\mathbf{x}) \\ &+ \lambda (w\_{l}(t) - w\_{l}(t-1)) \\ a\_{il}(t+1) &= a\_{il}(t) - \gamma \delta\_{l} \bigg( \frac{\cos(5q\_{il})e^{-\frac{1}{2}(q\_{il}^{2})}q\_{il}^{2} + 5q\_{il}\sin(5q\_{il})e^{-\frac{1}{2}(q\_{il}^{2})}}{a\_{il}} \bigg) \\ &+ \lambda (a\_{il}(t) - a\_{il}(t-1)) \\ \Rightarrow a\_{il}(t+1) &= a\_{il}(t) - \frac{\gamma(u(t) - r(t))\mu\_{l}(\mathbf{x})\,\omega\_{l}(q)}{\sum\_{l=1}^{n}\mu\_{l}(\mathbf{x})} \left( \frac{\cos(5q\_{il})e^{-\frac{1}{2}(q\_{il}^{2})}q\_{il}^{2} + 5q\_{il}\sin(5q\_{il})e^{-\frac{1}{2}(q\_{il}^{2})}}{a\_{il}} \right) \end{split} \tag{58}$$

$$+\lambda(a\_{il}(t) - a\_{il}(t-1))\tag{59}$$

$$b\_{il}(t+1) = b\_{il}(t) - \gamma \delta\_l \left(\frac{\cos(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}q\_{il} + \dots \sin(5q\_{il})e^{-\frac{1}{2}(q\_{il}^2)}}{a\_{il}}\right)$$

$$\begin{aligned} &+\lambda(b\_{\text{li}}(t)-b\_{\text{li}}(t-1)) \\ \Rightarrow b\_{\text{li}}(t+1) &= b\_{\text{li}}(t) - \frac{\gamma(u(t)-r(t))\mu\_{l}(\mathbf{x})\,\omega\_{l}(q)}{\sum\limits\_{l=1}^{n}\mu\_{l}(\mathbf{x})} \left(\frac{\cos(5q\_{\text{li}})e^{-\frac{1}{2}(q\_{\text{li}}^{2})}q\_{\text{li}} + \dots \sin(5q\_{\text{li}})e^{-\frac{1}{2}(q\_{\text{li}}^{2})}}{a\_{\text{il}}}\right) \end{aligned}$$

$$\begin{aligned} &+\lambda(b\_{il}(t) - b\_{il}(t-1)) \\ &= a \cdots (t) - \gamma \sum \left[ \left( \mu\_{\mathbf{u}(t)} - \mathbf{r}(t) \right) \underbrace{y\_{j} - u}\_{=} \mu\_{j} \sum \mathbf{g} \underbrace{\mathrm{sign}(\mathbf{x}\_{i} - \mathbf{g}\_{ij})}\_{=B} \right] \end{aligned} \tag{60}$$

$$g\_{\bar{i}\bar{j}}(t+1) = g\_{\bar{i}\bar{j}}(t) - \gamma \sum\_{\bar{j}} \left[ \left( u(t) - r(t) \right) . \frac{y\_{\bar{j}} - u}{\sum\_{\bar{j}} \mu\_{\bar{j}}} \frac{\mu\_{\bar{j}}}{\eta\_{\bar{j}}(\mathbf{x}\_{\bar{i}})} . 2 \frac{\operatorname{sign}(\mathbf{x}\_{\bar{i}} - \mathbf{g}\_{\bar{i}\bar{j}})}{\sigma\_{\bar{i}\bar{j}}} \right] \tag{61}$$

$$
\sigma\_{\rm ij}(t+1) = \sigma\_{\rm ij}(t) - \gamma \sum\_{j} \left[ \left( u(t) - r(t) . \right) \frac{y\_j - u}{\sum\_{j} \mu\_j} \cdot \frac{\mu\_j}{\eta\_{\rm j}(\mathbf{x}\_i)} \frac{1 - \eta\_j}{\sigma\_{\rm ij}} \right] \tag{62}
$$

Fig. 4. Overall control system design

*u* = [*u*1, *u*2, ··· , *us*]

functions. Let us refer;

and *z* = [*z*1, ··· , *zs*]

The control matrix is:

The disturbance matrix is:

The generic non-linear car model is,

is assumed available and *r* = *r*<sup>1</sup> + *r*<sup>2</sup> + ··· + *rp*.

*<sup>T</sup><sup>R</sup><sup>s</sup>* is the control input vector, *<sup>y</sup>* = [*y*1, ··· , *yp*]

*B*(*x*) =

*G*(*x*) =

*y*(*r*) = [*y*

⎡ ⎢ ⎣

⎡ ⎢ ⎣ . .

. .

(*r*1) <sup>1</sup> , *y* (*r*2) <sup>2</sup> , ··· , *y*

*A*(.) *Rp*×*p*; *B*(.) *Rp*×*<sup>s</sup>*

non-linear functions, *Bij*(*x*), *i* = 1, ··· , *p*; *j* = 1, ··· ,*s* are continuous non-linear control functions and *Gij*(*x*), *i* = 1, ··· , *p*; *j* = 1, ··· ,*s* are continuous non-linear disturbance

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 161

*b*11(*x*) ... *b*1*s*(*x*)

. ... .

*bp*1(*x*) ... *bps*(*x*)

*g*11(*x*) ... *g*1*s*(*x*)

. ... .

*gp*1(*x*) ... *gps*(*x*)

*y*˙ = *f*(*x*) + *B*(*x*).*u* + *G*(*x*).*z*

. .

> . .

> > (*rp* ) *<sup>p</sup>* ] *T*

*y*(*r*) = *A*(*x*) + *B*(*x*).*u* + *G*(*x*).*z* (65)

; *G*(.) *Rp*×*<sup>s</sup>*

*<sup>T</sup><sup>R</sup><sup>s</sup>* is the disturbance vector. *Ai*(*x*), *<sup>i</sup>* <sup>=</sup> 1, ··· , *<sup>p</sup>* are continuous

*<sup>A</sup>* = [*A*1(*x*) *<sup>A</sup>*2(*x*) ··· *Ap*(*x*)]*<sup>T</sup>* (64)

⎤ ⎥ ⎦ *p*×*s*

⎤ ⎥ ⎦ *p*×*s* *<sup>T</sup>R<sup>p</sup>* is the output vector

The above equations give the parameters updates for AFWNN-4.

#### **3. System modeling and design**

The proposed AFWNN structures have been applied to full car model with eight degree of freedom, being closer to reality, as shown in Figure 3. The eight degrees of freedom are the four wheels displacement (*Zf* ,*r*, *Zf* ,*l*, *Zr*,*r*, *Zr*,*l*), seat displacement '*Zs*', heave displacement '*Z*', pitch displacement '*θ*' and roll displacement '*φ*'. The car model comprises of only one sprung mass attached to the four unsprung masses at each corner. The sprung mass is allowed to have pitch, heave and roll and the unsprung masses are allowed to have heave only. For simplicity, all other motions are ignored for this model. The suspensions between the sprung mass and

Fig. 3. Full-Car Model

unsprung masses are modeled as non-linear viscous dampers and spring components and the tires are modeled as simple non-linear springs without damping elements. The actuator gives forces that determine by the displacement of the actuator between the sprung mass and the wheels. The dampers between the wheels and car body signify sources of conventional damping like friction among the mechanical components. The inputs of full-car model are four disturbances coming through the tires and the four outputs are the heave, pitch, seat, and roll displacement. For details of the dynamic model the reader is referred to (Rahmi, 2003). Figure 4 depicts the closed loop diagram of the feedback system. The input to the plant is the noisy output of controller. The controller parameters are adapted on the basis of calculated error which is the difference between the desired and actual output of the plant.

The inputs of the plant (full-car model) are four disturbances from the tire. The outputs are Seat, Heave, Pitch and Roll displacements. The states required for controller come from displacement sensors which measure the displacement states of four tires and one more sensor for measuring the displacement of seat. The adaptive control law uses control technique and adaptation mechanism to adapt the controller itself using proposed algorithms.The general class of nonlinear MIMO systems is described by;

$$y^{(r)} = A(\mathbf{x}) + \sum\_{i=1}^{p} \sum\_{j=1}^{s} B\_{ij}(\mathbf{x}) u\_j + \sum\_{i=1}^{p} \sum\_{j=1}^{s} G\_{ij}(\mathbf{x}) z\_j \tag{63}$$

Where, *x* = [*y*1, *y*˙1, ··· , *y* (*r*1−1) <sup>1</sup> , ··· , *yp*, *y*˙ *<sup>p</sup>*, ··· , *y* (*rp*−1) *<sup>p</sup>* ] *<sup>T</sup>�R<sup>r</sup>* is the overall state vector, which

#### Fig. 4. Overall control system design

is assumed available and *r* = *r*<sup>1</sup> + *r*<sup>2</sup> + ··· + *rp*.

*u* = [*u*1, *u*2, ··· , *us*] *<sup>T</sup><sup>R</sup><sup>s</sup>* is the control input vector, *<sup>y</sup>* = [*y*1, ··· , *yp*] *<sup>T</sup>R<sup>p</sup>* is the output vector and *z* = [*z*1, ··· , *zs*] *<sup>T</sup><sup>R</sup><sup>s</sup>* is the disturbance vector. *Ai*(*x*), *<sup>i</sup>* <sup>=</sup> 1, ··· , *<sup>p</sup>* are continuous non-linear functions, *Bij*(*x*), *i* = 1, ··· , *p*; *j* = 1, ··· ,*s* are continuous non-linear control functions and *Gij*(*x*), *i* = 1, ··· , *p*; *j* = 1, ··· ,*s* are continuous non-linear disturbance functions.

Let us refer;

14 Will-be-set-by-IN-TECH

The proposed AFWNN structures have been applied to full car model with eight degree of freedom, being closer to reality, as shown in Figure 3. The eight degrees of freedom are the four wheels displacement (*Zf* ,*r*, *Zf* ,*l*, *Zr*,*r*, *Zr*,*l*), seat displacement '*Zs*', heave displacement '*Z*', pitch displacement '*θ*' and roll displacement '*φ*'. The car model comprises of only one sprung mass attached to the four unsprung masses at each corner. The sprung mass is allowed to have pitch, heave and roll and the unsprung masses are allowed to have heave only. For simplicity, all other motions are ignored for this model. The suspensions between the sprung mass and

unsprung masses are modeled as non-linear viscous dampers and spring components and the tires are modeled as simple non-linear springs without damping elements. The actuator gives forces that determine by the displacement of the actuator between the sprung mass and the wheels. The dampers between the wheels and car body signify sources of conventional damping like friction among the mechanical components. The inputs of full-car model are four disturbances coming through the tires and the four outputs are the heave, pitch, seat, and roll displacement. For details of the dynamic model the reader is referred to (Rahmi, 2003). Figure 4 depicts the closed loop diagram of the feedback system. The input to the plant is the noisy output of controller. The controller parameters are adapted on the basis of calculated error which is the difference between the desired and actual output of the plant. The inputs of the plant (full-car model) are four disturbances from the tire. The outputs are Seat, Heave, Pitch and Roll displacements. The states required for controller come from displacement sensors which measure the displacement states of four tires and one more sensor for measuring the displacement of seat. The adaptive control law uses control technique and adaptation mechanism to adapt the controller itself using proposed algorithms.The general

The above equations give the parameters updates for AFWNN-4.

**3. System modeling and design**

Fig. 3. Full-Car Model

Where, *x* = [*y*1, *y*˙1, ··· , *y*

class of nonlinear MIMO systems is described by;

*y*(*r*) = *A*(*x*) +

(*r*1−1)

*p* ∑ *i*=1

<sup>1</sup> , ··· , *yp*, *y*˙ *<sup>p</sup>*, ··· , *y*

*s* ∑ *j*=1

*Bij*(*x*)*uj* +

(*rp*−1) *<sup>p</sup>* ]

*p* ∑ *i*=1

*s* ∑ *j*=1

*Gij*(*x*)*zj* (63)

*<sup>T</sup>�R<sup>r</sup>* is the overall state vector, which

$$A = \begin{bmatrix} A\_1(\mathbf{x}) \ A\_2(\mathbf{x}) \ \cdots \ A\_p(\mathbf{x}) \end{bmatrix}^T \tag{64}$$

The control matrix is:

$$B(\mathbf{x}) = \begin{bmatrix} b\_{11}(\mathbf{x}) \dots \ b\_{1s}(\mathbf{x}) \\ \vdots & \ddots & \vdots \\ b\_{p1}(\mathbf{x}) \dots \ b\_{ps}(\mathbf{x}) \end{bmatrix}\_{p \times s}$$

The disturbance matrix is:

$$\begin{aligned} G(\mathbf{x}) &= \begin{bmatrix} g\_{11}(\mathbf{x}) \dots \ g\_{1s}(\mathbf{x}) \\ \vdots & \ddots & \vdots \\ g\_{p1}(\mathbf{x}) \dots \ g\_{ps}(\mathbf{x}) \end{bmatrix}\_{p \times s} \\\\ y^{(r)} &= [y\_1^{(r\_1)}, y\_2^{(r\_2)}, \dots, y\_p^{(r\_p)}]^T \\\\ y^{(r)} &= A(\mathbf{x}) + B(\mathbf{x}).u + G(\mathbf{x}).z \\\\ A(.) \,\epsilon \, R^{p \times p}; \quad B(.) \,\epsilon \, R^{p \times s}; \quad G(.) \,\epsilon \, R^{p \times s} \\\\ \text{model is} \end{aligned} \tag{65}$$

The generic non-linear car model is,

$$\dot{y} = f(\mathbf{x}) + \mathcal{B}(\mathbf{x})\,\mu + G(\mathbf{x})\,z$$

Constants Description Values Units *k <sup>f</sup>* 1, *k <sup>f</sup>* <sup>2</sup> Front-left and Front-right suspension stuffiness, respectively. 15000 N/m *kr*1, *kr*<sup>2</sup> Rear-left and rear-right suspension stuffiness, respectively. 17000 N/m *ks* Seat spring Constant 15000 N/m *cs* Seat damping Constant 15000 N/m

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 163

Front-left, Front-right, rear-right and rear-left tire damping,

Table 1. Vehicle Suspension Parameters

Table 2. Learning rates '*γ*' for controls

*cs*<sup>1</sup> − *cs*<sup>4</sup> respectively. 2500 N.sec/m Front-left, Front-right, rear-right and rear-left tire suspension, *kt*<sup>1</sup> − *kt*<sup>4</sup> respectively. 250000 N/m *a* Distance between front axle suspension and C.O.G. 1.2 m *b* Distance between rear axle suspension and C.O.G. 1.4 m *Xs* Horizontal distance of seat from C.O.G. 0.3 m *Ys* Vertical distance of seat from C.O.G. 0.25 m *Mf* ,*l*, *Mf* ,*<sup>r</sup>* Front-left and Front-right unsprung mass, respectively. 25 kg *Mr*,*l*, *Mr*,*<sup>r</sup>* Rear-left and rear-right unsprung mass, respectively. 45 kg *Ms* Seat Mass 90 kg *M* Vehicle body mass 1100 kg *Ix* Moment of inertia for pitch 1848 *kg*.*m*<sup>2</sup> *Iy* Moment of inertia for roll 550 *kg*.*m*<sup>2</sup> Cshy1 Shyhook damper constant -2500 N.sec/m

and AFWNNC-2 use Mexican hat as wavelet in the consequent part and gaussian and triangular as membership function in the antecedent, respectively. AFWNNC-3 and AFWNNC-4 use Morlet as wavelet in the consequent part and gaussian and triangular as membership function in the antecedent, respectively. Two rules each having two membership functions have been used for simulation. For each AFWNN, 18 parameters have been adapted being the mean and variance of the antecedent part and translation, dilation and weights of the consequent part. Three types of road profiles have been examined to check the robustness

Sr. No. Control Algo. Seat Front Rear

 APID 0.9 1 0.6 0.5 0.8 AFWNN-1 0.001 0.4 0.09 0.7 0.2 AFWNN-2 0.0054 0.08 0.09 0.08 0.09 AFWNN-3 0.003 0.08 0.007 0.009 0.008 AFWNN-4 0.003 0.0009 0.001 0.004 0.006

of the applied algorithms. These road profiles have been used in context of roll, pitch, heave and seat displacement and acceleration. Four controllers have been applied to each car tire and one has been taken for seat. The learning rates for each controller have been shown in Table 2. These values have been set for learning rates based on hit-and-trial keeping in view the fact that a positive change in the error rate leads to increase the value of '*γ*' and vice versa.

For simplicity of implementation the moment term has been neglected.

Left Right Left Right

*y* = *h*(*x*)

Where, *f*(*x*) *R*(16×16), *B*(*x*) *R*(16×4), *G*(*x*) *R*(16×4), state vector *x R*(16×1), *u R*(4×1) and *z R*(4×1).

These matrices can be shown in state-space form, with state vector *x*, represented in row matrix form.

$$\begin{aligned} f(\mathbf{x}) &= \begin{bmatrix} A\_1(\mathbf{x}) & A\_2(\mathbf{x}) & A\_3(\mathbf{x}) & \dots & A\_{16}(\mathbf{x}) \end{bmatrix} \\ \mathbf{x} &= \begin{bmatrix} \mathbf{x}\_1 & \mathbf{x}\_2 & \mathbf{x}\_3 & \dots & \mathbf{x}\_{16} \end{bmatrix}^T \end{aligned}$$

*A*1(*x*) to *A*8(*x*) are velocity states and *A*9(*x*) to *A*16(*x*) are acceleration states of four tires, seat, heave, pitch and roll.

The disturbance inputs for each tire individually are represented in the form of *z* matrix.

$$z = \begin{bmatrix} z\_1 & z\_2 & z\_3 & z\_4 \end{bmatrix}^T$$

*zn* are *n* disturbances applied to full-car model. *un* are *n* control inputs to full-car model, so to regulate the car model disturbances. *yn* are *n* states of car. *rn* are *n* desired outputs for the controller to achieve.

Each **controller** in this work has two inputs. One of the inputs is '*rn*' and delay of it is given to second input . The *yn* states are fed to the controller as an error, so to adapt the update adaptation law for the desired regulation. Based on this error the adaptation law is formulated using AFWNN-1, AFWNN-2, AFWNN-3 and AFWNN-4. The algorithms develop a back-propagation algorithm for training the controller to achieve the desire performance.

In this work for the full-car model **four states of tires** are used by the four controllers as an error to adapt the adaptation law. As the purpose of controller is to regulate the disturbances so *rn*'s are zero, the second input of controllers is a delayed version of first input. The adaptation law of the controller provides the control inputs *u*1, *u*2, *u*<sup>3</sup> and *u*<sup>4</sup> to plant so as to regulate the plant. The four disturbances *z*1, *z*2, *z*<sup>3</sup> and *z*<sup>4</sup> are coming from road through tires into suspension system and to the body of the vehicle.

**Two cases** have been considered. In the **first case**, only the states of the four tires *y*1, *y*2, *y*<sup>3</sup> and *y*<sup>4</sup> displacement are used as an error to the controller. Which develops the control law according to that error. These control inputs *u*1, *u*2, *u*<sup>3</sup> and *u*<sup>4</sup> are provided to the plant from each controller (placed on each tire) to achieve the desired performance of the plant (full-car model) i.e. both better passenger comfort (better seat and heave displacement) and better vehicle stability (better heave, pitch and roll displacement). In the **second case**, an additional controller is applied under the driver seat to improve the passenger comfort. In this case, another state *y*<sup>5</sup> is used as an error input to the controller. This additional control input will help in reducing the disturbance effect and improving the passenger comfort.

Table 1 gives the description of different constants and their respective values used for simulation.

#### **4. Simulation results and discussion**

Four different types of fuzzy wavelet neural network control techniques in addition to the APID and semi-active control have been applied to full car suspension model. AFWNNC-1


Table 1. Vehicle Suspension Parameters

16 Will-be-set-by-IN-TECH

*y* = *h*(*x*) Where, *f*(*x*) *R*(16×16), *B*(*x*) *R*(16×4), *G*(*x*) *R*(16×4), state vector *x R*(16×1), *u R*(4×1) and

These matrices can be shown in state-space form, with state vector *x*, represented in row

*f*(*x*)=[*A*1(*x*) *A*2(*x*) *A*3(*x*) ... *A*16(*x*)]

*A*1(*x*) to *A*8(*x*) are velocity states and *A*9(*x*) to *A*16(*x*) are acceleration states of four tires, seat,

The disturbance inputs for each tire individually are represented in the form of *z* matrix.

*z* = [*z*<sup>1</sup> *z*<sup>2</sup> *z*<sup>3</sup> *z*4]

*zn* are *n* disturbances applied to full-car model. *un* are *n* control inputs to full-car model, so to regulate the car model disturbances. *yn* are *n* states of car. *rn* are *n* desired outputs for the

Each **controller** in this work has two inputs. One of the inputs is '*rn*' and delay of it is given to second input . The *yn* states are fed to the controller as an error, so to adapt the update adaptation law for the desired regulation. Based on this error the adaptation law is formulated using AFWNN-1, AFWNN-2, AFWNN-3 and AFWNN-4. The algorithms develop a back-propagation algorithm for training the controller to achieve the desire performance. In this work for the full-car model **four states of tires** are used by the four controllers as an error to adapt the adaptation law. As the purpose of controller is to regulate the disturbances so *rn*'s are zero, the second input of controllers is a delayed version of first input. The adaptation law of the controller provides the control inputs *u*1, *u*2, *u*<sup>3</sup> and *u*<sup>4</sup> to plant so as to regulate the plant. The four disturbances *z*1, *z*2, *z*<sup>3</sup> and *z*<sup>4</sup> are coming from road through

**Two cases** have been considered. In the **first case**, only the states of the four tires *y*1, *y*2, *y*<sup>3</sup> and *y*<sup>4</sup> displacement are used as an error to the controller. Which develops the control law according to that error. These control inputs *u*1, *u*2, *u*<sup>3</sup> and *u*<sup>4</sup> are provided to the plant from each controller (placed on each tire) to achieve the desired performance of the plant (full-car model) i.e. both better passenger comfort (better seat and heave displacement) and better vehicle stability (better heave, pitch and roll displacement). In the **second case**, an additional controller is applied under the driver seat to improve the passenger comfort. In this case, another state *y*<sup>5</sup> is used as an error input to the controller. This additional control input will

Table 1 gives the description of different constants and their respective values used for

Four different types of fuzzy wavelet neural network control techniques in addition to the APID and semi-active control have been applied to full car suspension model. AFWNNC-1

help in reducing the disturbance effect and improving the passenger comfort.

tires into suspension system and to the body of the vehicle.

*T*

*T*

*x* = [*x*<sup>1</sup> *x*<sup>2</sup> *x*<sup>3</sup> ... *x*16]

*z R*(4×1).

matrix form.

heave, pitch and roll.

controller to achieve.

simulation.

**4. Simulation results and discussion**

and AFWNNC-2 use Mexican hat as wavelet in the consequent part and gaussian and triangular as membership function in the antecedent, respectively. AFWNNC-3 and AFWNNC-4 use Morlet as wavelet in the consequent part and gaussian and triangular as membership function in the antecedent, respectively. Two rules each having two membership functions have been used for simulation. For each AFWNN, 18 parameters have been adapted being the mean and variance of the antecedent part and translation, dilation and weights of the consequent part. Three types of road profiles have been examined to check the robustness


Table 2. Learning rates '*γ*' for controls

of the applied algorithms. These road profiles have been used in context of roll, pitch, heave and seat displacement and acceleration. Four controllers have been applied to each car tire and one has been taken for seat. The learning rates for each controller have been shown in Table 2. These values have been set for learning rates based on hit-and-trial keeping in view the fact that a positive change in the error rate leads to increase the value of '*γ*' and vice versa. For simplicity of implementation the moment term has been neglected.

(a) Heave (b) Pitch

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 165

(c) Roll (d) Seat

(e) Front left tire (f) Front right tire

(g) Rear left tire (h) Rear right tire

(i) Seat

Fig. 6. (a) Heave amplitude (b) Pitch amplitude (c) Roll amplitude (d) Seat amplitude (e)-(i) Update parameters for antecedent and consequent part of AFWNN-4 for all five controllers

The performance index used for evaluation of different algorithms is given by,

$$I = \frac{1}{2} \int\_0^T (Z\_P^T Q Z\_P) dt\tag{66}$$

where, '*Zp*' is the vector for displacement or acceleration, 'Q' is the identity matrix. The Root Mean Square (RMS) value for displacement and acceleration of heave, pitch, roll and seat has been calculated by,

$$z\_{\text{disp.}}^{\text{rms}} = \sqrt{\frac{1}{T} \int\_{t=0}^{T} [h(t)]^2} \tag{67}$$

$$
\ddot{z}\_{\rm acc.}^{rms} = \sqrt{\frac{1}{T} \int\_{t=0}^{T} [\ddot{h}(t)]^2} \tag{68}
$$

Figure 5 shows different road profiles used for simulation.

Fig. 5. (a) Road profile-1 (b) Road profile-2 (c) Road profile-3

18 Will-be-set-by-IN-TECH

where, '*Zp*' is the vector for displacement or acceleration, 'Q' is the identity matrix. The Root Mean Square (RMS) value for displacement and acceleration of heave, pitch, roll and seat has

> 1 *T T t*=0

 1 *T T t*=0 [¨

(a) Road profile-1

(b) Road profile-2 for front and rear left tires (c) Road profile-2 for front and rear right tires

(d) Road profile-3

Fig. 5. (a) Road profile-1 (b) Road profile-2 (c) Road profile-3

*<sup>P</sup>QZP*)*dt* (66)

[*h*(*t*)]<sup>2</sup> (67)

*h*(*t*)]<sup>2</sup> (68)

The performance index used for evaluation of different algorithms is given by,

*<sup>I</sup>* <sup>=</sup> <sup>1</sup> 2 *T* 0 (*Z<sup>T</sup>*

*zrms disp*. =

> *z*¨ *rms acc*. =

Figure 5 shows different road profiles used for simulation.

been calculated by,

Fig. 6. (a) Heave amplitude (b) Pitch amplitude (c) Roll amplitude (d) Seat amplitude (e)-(i) Update parameters for antecedent and consequent part of AFWNN-4 for all five controllers

Road Profile Control Algo. Performance Index RMS % Improvement w.r.t.

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 167

1 AFWNN-1 4.6587 0.0158 3.0524 75 65

2 AFWNN-1 0.7863 0.0021 1.2540 54 10

3 AFWNN-1 1.5545 0.01 1.7632 54 40

4 AFWNN-1 0.7966 0.017 1.2621 70 49

Passive 6.03515 0.0404 3.476 - - Semi-active 5.3037 0.03729 3.2567 - - APID 4.92835 0.0166 3.1395 70 62

AFWNN-2 4.3854 0.0148 2.965 77 65 AFWNN-3 1.9529 0.00403 1.9763 93 89 AFWNN-4 1.8006 0.004 1.8977 94 93

Passive 1.6822 0.00545 1.8342 - - Semi-active 1.4578 0.00443 1.7075 - - APID 1.3194 0.00263 1.6244 53 05

AFWNN-2 0.7213 0.0020 1.2011 57 15 AFWNN-3 0.4600 0.0031 0.9590 57 25 AFWNN-4 0.2494 0.0022 0.7063 62 35

Passive 3.6786 0.0308 2.7122 - - Semi-active 3.5407 0.0274 2.6610 - - APID 2.9947 0.0094 2.4473 50 38

AFWNN-2 1.1381 0.01004 1.2011 69 62 AFWNN-3 0.9288 0.005 1.3629 80 73 AFWNN-4 0.7834 0.0041 1.2517 84 80

Passive 3.8449 0.04695 2.7726 - - Semi-active 1.8379 0.04101 1.9168 - - APID 0.8508 0.02 1.3043 55 28

AFWNN-2 0.6492 0.031 1.1319 72 51 AFWNN-3 0.066 0.0052 0.3631 87 77 AFWNN-4 0.0295 0.0039 0.2428 88 81

> *Aisin*(Ω*is* − Ψ*i*) 0 ≤ *t* ≤ 16 0 otherwise

Where, value of '*Ai*' is the road amplitude, 'Ω*i*' is the number of waves and 'Ψ*i*' is the phase

The control problem is that the suspension travel should be | *z* | less than | *z* | from the amplitude of disturbance i.e., ±0.15*m*. The maximum displacement of the road profile is

(71)

Heave

Roll

Pitch

Seat

**4.3 Road profile-3**

±0.15*m*.

Table 3. Performance Comparison for road profile-1

Road profile-3 is white noise as shown in Figure 5(a).

angle, *i* = 1, 2, ..., *N* ranging from 0 to 2*π*.

*z*3(*t*) =

⎧ ⎪⎨

*N* ∑ *i*=1

⎪⎩

Disp. Acc. Passive Semi-active

#### **4.1 Road profile-1**

Road profile-1 involves one pothole and one bump, each having duration of one second with a time delay of 8 secs., for front and rear tires. Mathematically, road profile-1 is given by;

$$z\_1(t) = \begin{cases} \text{-0.15} & 1 \le t \le 2 \text{ and } 4 \le t \le 5\\ \text{0.15} & 9 \le t \le 10 \text{ and } 12 \le t \le 13\\ 0 & \text{otherwise} \end{cases} \tag{69}$$

i.e., the road profile contains a pothole and a bump of amplitudes −0.15*m* and 0.15*m*, respectively. This road profile is helpful to calculate heave of a vehicle. Figure 5(a) depicts the road profile−1. Figures 6(a)-(d) show the regulation results for heave, roll, pitch and seat displacement for active suspension as compared to passive and semi-active suspension. It is clear from the figures that there is improvement in the results for active suspension. The settling time has been reduced and the steady state response is improved. In case of heave and seat the passive control approaches the rattle space limits whereas AFWNN-4 has optimal results for all the four cases showing the least variation from steady state.

In passive and semi-active suspension suspension, the maximum values of displacements for heave is 0.106*m* and 0.088*m*, for roll 0.016*m* and 0.009*m*, for pitch is 0.075*m* and 0.061*m* and for seat is 0.15*m* and 0.11, respectively. Due to high nonlinear nature of AFWNN-4 these values get improved as 0.004*m*, 0.006*m*, 0.012*m* and 0.02 for heave, roll pitch and seat, respectively.

Table3 shows the results for percent improvement and RMS values for displacement and acceleration, for road profile-1. It can be seen that maximum improvement has been achieved in case of heave with AFWNN-4. Figures 6(e)-(i) show the antecedent and consequent parameters variation for AFWNN-4 for all the five controls. Parameters variation for front and rear right tires is large whereas front and rear left tire has low parameters variation. It was found that the control effort by front and right tires was greater as compared to seat and the left side tires controls.

#### **4.2 Road profile-2**

Road profile-2 has been taken as two potholes of different amplitudes as shown in Figures 5(b)-(c). The road profile−2 is given as follows:

$$z\_2(t) = \begin{cases} \text{-0.15} & 1 \le t \le 2 \text{ and } 9 \le t \le 10\\ \text{-0.10} & 4 \le t \le 5 \text{ and } 12 \le t \le 13\\ 0 & \text{otherwise} \end{cases} \tag{70}$$

Road profile-2 involves two different potholes of amplitudes −0.15*m* and −0.10*m* for front and rear left and rear and front right, respectively. This road profile is very helpful for the calculation of pitch and roll of the vehicle.

Figures 7(a)-(d) reveal that APID shows satisfactory results whereas the result are very good in case of AFWNN-4. The maximum improvement has been found in case of roll for this road profile, which corresponds to the control of vehicle around horizontal axis. Figures 7(e)-(i) give the update parameters results for AFWNN-4 showing large variations for rear left and rear right tires. Table 4 shows the results for road profile-2 in terms of percent improvement and RMS values of displacement and acceleration. The passive and semi-active suspension show poor performance in terms of passenger comfort and vehicle stability.


Table 3. Performance Comparison for road profile-1

#### **4.3 Road profile-3**

20 Will-be-set-by-IN-TECH

Road profile-1 involves one pothole and one bump, each having duration of one second with a time delay of 8 secs., for front and rear tires. Mathematically, road profile-1 is given by;

i.e., the road profile contains a pothole and a bump of amplitudes −0.15*m* and 0.15*m*, respectively. This road profile is helpful to calculate heave of a vehicle. Figure 5(a) depicts the road profile−1. Figures 6(a)-(d) show the regulation results for heave, roll, pitch and seat displacement for active suspension as compared to passive and semi-active suspension. It is clear from the figures that there is improvement in the results for active suspension. The settling time has been reduced and the steady state response is improved. In case of heave and seat the passive control approaches the rattle space limits whereas AFWNN-4 has optimal

In passive and semi-active suspension suspension, the maximum values of displacements for heave is 0.106*m* and 0.088*m*, for roll 0.016*m* and 0.009*m*, for pitch is 0.075*m* and 0.061*m* and for seat is 0.15*m* and 0.11, respectively. Due to high nonlinear nature of AFWNN-4 these values get improved as 0.004*m*, 0.006*m*, 0.012*m* and 0.02 for heave, roll pitch and seat, respectively. Table3 shows the results for percent improvement and RMS values for displacement and acceleration, for road profile-1. It can be seen that maximum improvement has been achieved in case of heave with AFWNN-4. Figures 6(e)-(i) show the antecedent and consequent parameters variation for AFWNN-4 for all the five controls. Parameters variation for front and rear right tires is large whereas front and rear left tire has low parameters variation. It was found that the control effort by front and right tires was greater as compared to seat and

Road profile-2 has been taken as two potholes of different amplitudes as shown in Figures

Road profile-2 involves two different potholes of amplitudes −0.15*m* and −0.10*m* for front and rear left and rear and front right, respectively. This road profile is very helpful for the

Figures 7(a)-(d) reveal that APID shows satisfactory results whereas the result are very good in case of AFWNN-4. The maximum improvement has been found in case of roll for this road profile, which corresponds to the control of vehicle around horizontal axis. Figures 7(e)-(i) give the update parameters results for AFWNN-4 showing large variations for rear left and rear right tires. Table 4 shows the results for road profile-2 in terms of percent improvement and RMS values of displacement and acceleration. The passive and semi-active suspension

0 otherwise


0 otherwise


(69)

(70)

*z*1(*t*) =

⎧ ⎪⎨

⎪⎩

results for all the four cases showing the least variation from steady state.

**4.1 Road profile-1**

the left side tires controls.

5(b)-(c). The road profile−2 is given as follows:

calculation of pitch and roll of the vehicle.

*z*2(*t*) =

⎧ ⎪⎨

⎪⎩

show poor performance in terms of passenger comfort and vehicle stability.

**4.2 Road profile-2**

Road profile-3 is white noise as shown in Figure 5(a).

$$z\_3(t) = \begin{cases} \sum\_{i=1}^{N} A\_i \sin(\Omega\_i s - \Psi\_i) & 0 \le t \le 16\\ 0 & \text{otherwise} \end{cases} \tag{71}$$

Where, value of '*Ai*' is the road amplitude, 'Ω*i*' is the number of waves and 'Ψ*i*' is the phase angle, *i* = 1, 2, ..., *N* ranging from 0 to 2*π*.

The control problem is that the suspension travel should be | *z* | less than | *z* | from the amplitude of disturbance i.e., ±0.15*m*. The maximum displacement of the road profile is ±0.15*m*.

(a) Heave (b) Pitch

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 169

(c) Roll (d) Seat

(e) Front left tire (f) Front right tire

(g) Rear left tire (h) Rear right tire

(i) Seat

Fig. 7. (a) Heave amplitude (b) Pitch amplitude (c) Roll amplitude (d) Seat amplitude (e)-(i) Update parameters for antecedent and consequent part of AFWNN-4 for all five controllers


Table 4. Performance comparison for road profile-2

The time delay between front and rear wheels is given by;

$$\delta(t) = \frac{(s\_1 + s\_2)}{V} \tag{72}$$

Where, *s*<sup>1</sup> = 1.2*m* and *s*<sup>2</sup> = 1.4*m* are the values of distance between front and rear wheels and '*V*' is the vehicle velocity. Figures 8(a)-(d) show the results for displacement for different car parameters for each algorithm. There is a performance degradation in case of AFWNN-2 for pitch. Figures 8(e)-(i) show the update parameters for consequent and antecedent part of AFWNN-4. Table 5 shows the performance comparison for road profile-3 for different parameters. The best results have been obtained in case of AFWNN-4 for seat in this case. The minimum displacement for seat correspond to the passenger comfort which shows that AFWNN-4 gives optimal results for passenger comfort for comparatively rough road profiles. It can be seen that the performance difference between AFWNN-1 and AFWNN-2 is small as compared to that of AFWNN-1 and AFWNN-3 or AFWNN-4 which shows that incorporation of Morlet wavelet has improved the performance consistency, significantly.

22 Will-be-set-by-IN-TECH

Road Profile Control Algo. Performance Index RMS % Improvement w.r.t.

1 AFWNN-1 2.9745 0.01397 2.439 60 47

2 AFWNN-1 0.8991 0.0022 1.3410 57 46

3 AFWNN-1 1.8417 0.0080 1.9190 50 10

4 AFWNN-1 0.7014 0.1631 1.1731 69 40

*<sup>δ</sup>*(*t*) = (*s*<sup>1</sup> <sup>+</sup> *<sup>s</sup>*2)

Where, *s*<sup>1</sup> = 1.2*m* and *s*<sup>2</sup> = 1.4*m* are the values of distance between front and rear wheels and '*V*' is the vehicle velocity. Figures 8(a)-(d) show the results for displacement for different car parameters for each algorithm. There is a performance degradation in case of AFWNN-2 for pitch. Figures 8(e)-(i) show the update parameters for consequent and antecedent part of AFWNN-4. Table 5 shows the performance comparison for road profile-3 for different parameters. The best results have been obtained in case of AFWNN-4 for seat in this case. The minimum displacement for seat correspond to the passenger comfort which shows that AFWNN-4 gives optimal results for passenger comfort for comparatively rough road profiles. It can be seen that the performance difference between AFWNN-1 and AFWNN-2 is small as compared to that of AFWNN-1 and AFWNN-3 or AFWNN-4 which shows that incorporation

of Morlet wavelet has improved the performance consistency, significantly.

Passive 4.2056 0.0339 2.9 - - Semi-active 3.8635 0.09855 2.778 - - APID 3.1469 0.01542 2.5087 57 36

AFWNN-2 3.0190 0.0144 2.453 80 72 AFWNN-3 1.2864 0.0036 1.604 83 78 AFWNN-4 1.1303 0.0040 1.503 87 82

Passive 1.7504 0.0099 1.8710 - - Semi-active 1.5832 0.00679 1.7794 - - APID 1.2140 0.00438 1.5582 36 25

AFWNN-2 0.7176 0.0020 1.1980 63 46 AFWNN-3 0.5925 0.0024 1.0886 74 66 AFWNN-4 0.5011 0.0019 1.0010 95 92

Passive 2.5485 0.0257 2.2575 - - Semi-active 2.2184 0.0193 2.1063 - - APID 1.9173 0.0119 1.9582 60 53

AFWNN-2 1.6864 0.0067 1.8365 60 52 AFWNN-3 0.8484 0.0031 1.3026 78 74 AFWNN-4 0.8831 0.0024 1.3290 81 78

Passive 2.6620 0.0391 2.3011 - - Semi-active 1.8148 0.03065 1.9049 - - APID 0.7233 0.1850 1.2026 49 30

AFWNN-2 0.7325 0.1432 1.2019 69 43 AFWNN-3 0.0533 0.0046 0.3265 83 69 AFWNN-4 0.0317 0.0024 0.2516 85 72

Heave

Roll

Pitch

Seat

Table 4. Performance comparison for road profile-2

The time delay between front and rear wheels is given by;

Disp. Acc. Passive Semi-active

*<sup>V</sup>* (72)

Fig. 7. (a) Heave amplitude (b) Pitch amplitude (c) Roll amplitude (d) Seat amplitude (e)-(i) Update parameters for antecedent and consequent part of AFWNN-4 for all five controllers

(a) Heave (b) Pitch

Adaptive Fuzzy Wavelet NN Control Strategy for Full Car Suspension System 171

(c) Roll (d) Seat

(e) Front left tire (f) Front right tire

(g) Rear left tire (h) Rear right tire

(i) Seat

Fig. 8. (a) Heave amplitude (b) Pitch amplitude (c) Roll amplitude (d) Seat amplitude (e)-(i) Update parameters for antecedent and consequent part of AFWNN-4 for all five controllers



24 Will-be-set-by-IN-TECH

Parameters Control Algo. Performance Index RMS % Improvement w.r.t.

1 FWNN-1 12.5589 0.02607 5.0117 64 55

2 FWNN-1 0.1497 0.0190 4.5010 65 60

3 FWNN-1 1.7559 0.0045 1.8760 59 56

4 FWNN-1 9.3490 0.02607 0.3142 72 45

Passive 34.0666 0.0682 8.254 - - Semi-active 33.4707 0.06684 8.1815 - - APID 14.2367 0.0239 5.336 64 53

FWNN-2 12.8224 0.0262 5.064 65 64 FWNN-3 8.0526 0.0318 4.013 79 73 FWNN-4 5.1264 0.00956 3.202 87 83

Passive 25.3794 0.0495 7.1243 - - Semi-active 18.4609 0.0372 6.0762 - - APID 13.0721 0.0223 5.1131 52 45

FWNN-2 7.6561 0.1903 3.9085 70 66 FWNN-3 5.0197 0.0430 3.1682 81 78 FWNN-4 3.9790 0.0219 2.8209 88 85

Passive 3.8752 0.0216 2.7839 - - Semi-active 3.1087 0.0192 2.4934 - - APID 2.1505 0.0073 2.0739 40 44

FWNN-2 2.0279 0.0060 2.0739 50 45 FWNN-3 1.4126 0.166 1.6808 68 64 FWNN-4 0.7421 0.1403 1.2176 76 73

Passive 84.3201 0.1233 12.985 - - Semi-active 60.0559 0.1098 10.9590 - - APID 14.4606 0.0486 0.0486 65 35

FWNN-2 6.3870 0.0262 0.0221 73 46 FWNN-3 3.8223 0.0318 0.0278 80 67 FWNN-4 2.2046 0.0102 2.0998 90 92

Heave

Roll

Pitch

Seat

Table 5. Performance comparison for road profile-3

Disp. Acc. Passive Semi-active

Fig. 8. (a) Heave amplitude (b) Pitch amplitude (c) Roll amplitude (d) Seat amplitude (e)-(i) Update parameters for antecedent and consequent part of AFWNN-4 for all five controllers

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#### **5. Conclusion**

The detailed mathematical modeling of different adaptive softcomputing techniques have been developed and successfully applied to a full car model. The robustness of the presented techniques has been proved on the basis of different performance indices. Unlike, the conventional PID, the proposed algorithms have been compared with each other and APID controller. The simulation results and their analysis reveal that proposed AFWNNC gives better ride comfort and vehicle handling as compared to passive or semi-active and APID control. The performance of the active suspension has been optimized in terms of seat, heave, pitch and roll displacement and acceleration. The results show that AFWNNC-4 gives optimal performance for all rotational and translational motions of the vehicle persevering the passenger comfortability.

#### **6. References**


26 Will-be-set-by-IN-TECH

The detailed mathematical modeling of different adaptive softcomputing techniques have been developed and successfully applied to a full car model. The robustness of the presented techniques has been proved on the basis of different performance indices. Unlike, the conventional PID, the proposed algorithms have been compared with each other and APID controller. The simulation results and their analysis reveal that proposed AFWNNC gives better ride comfort and vehicle handling as compared to passive or semi-active and APID control. The performance of the active suspension has been optimized in terms of seat, heave, pitch and roll displacement and acceleration. The results show that AFWNNC-4 gives optimal performance for all rotational and translational motions of the vehicle persevering the

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**5. Conclusion**

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**9** 

*India* 

**Condition Ranking and Rating of Bridges Using Fuzzy Logic** 

*CSIR-Structural Engineering Research Centre,* 

*CSIR Complex, TTTI Post, Taramani,* 

Saptarshi Sasmal, K. Ramanjaneyulu and Nagesh R. Iyer

Bridges are the crucial components of highway networks. In recent years, there has been growing awareness about the problems associated with the existing old bridges. Many of the existing bridges in service today were designed for less traffic, smaller vehicles, slower speeds and lighter traffic. Hence, they have become inadequate according to the current loading standards/codes of practice for design of highway bridges. Even in the case of newer bridges, deterioration caused by unforeseen service condition, adverse environmental actions and inadequate maintenance is causing great concern to bridge engineers. Bridge authority has the responsibility to maintain its bridges in a safe condition. To ensure safe and durable service, it is usual to perform periodic in-situ inspections. These inspections involve visual observations, non-destructive testing and partial destructive testing. The data collected from site and processed in laboratory, would be used to decide about suitable repair, strengthening or demolition of existing bridges. It is also evident that engineers and decision makers have to deal with large number of deficient bridges in years to come and it will be extremely demanding to decide the most deserving one to allot fund for timely retrofitting. Further, it is necessary to formulate a systematic method to assess the present and future needs of the existing bridges which would help the decision makers in

identifying the most deserving bridges for improvement during a given period.

In view of this, several countries have initiated development of bridge management systems for assisting their decision makers in finding optimal strategies for maintenance, rehabilitation and replacement of bridges. Furthermore, it also has to ensure value for money by carrying out preventive work at appropriate time so that future maintenance needs are also kept at a minimum level. In a broader sense, the funding body has to consider the justification and priority for money to be spent on a multitude of expenditure areas. Decision makers and/or society at large should be able to choose whether to spend money on rehabilitating a bridge or to demolish it. The bridge engineers and the policy makers are being increasingly pressed to justify the funding order proposed to maintain the bridges. It shows the importance of an exclusive bridge management system. Bridge management is a rational and systematic approach for organising and carrying out the activities related to planning, design, construction, maintenance, rehabilitation and replacement of bridges.

**1. Introduction** 

	- URL: *http://white-smoke.wetpaint.com/page/Heave,+Pitch,+Roll,+Warp+and+Yaw*

### **Condition Ranking and Rating of Bridges Using Fuzzy Logic**

Saptarshi Sasmal, K. Ramanjaneyulu and Nagesh R. Iyer *CSIR-Structural Engineering Research Centre, CSIR Complex, TTTI Post, Taramani, India* 

#### **1. Introduction**

28 Will-be-set-by-IN-TECH

174 Fuzzy Logic – Emerging Technologies and Applications

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Yilmaz, S. & Oysal, Y. (2010). Fuzzy wavelet neural network models for prediction

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Yoshimura, T., Nakaminami, K., Kurimoto, M. & Hino, J. (1999). Active suspension of

Yue, C., Butsuen, T. & Hedrick, J. K. (1988). Alternative control laws for automotive active suspensions, *American Control Conference*, IEEE, Atlanta, USA, pp. 2373–2378. Zhang, J., Walter, G. G., Miao, Y. & Lee, W. N. W. (1995). Wavelet neural networks for function

Zhang, Q. (1997). Using wavelet networks in nonparametric estimation, *IEEE Trans. Neural*

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and identification of dynamical systems, *IEEE Transactions on Neural Networks*

system of a quarter car model using the concept of sliding mode control, *Journal of*

passengers cars using linear and fuzzy-logic controls, *Control Engineering Practice*

decomposition methods, *Journal of Sound and Vibration* 285(3): 571–583. Thompson, A. G. & Pearce, C. E. M. (1998). Physically realizable feedback controls for a

for vehicle active suspension systems, *IEEE Transactions on Industrial Electronics*

control forces in a half-car model with preview active suspension using spectral

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vehicles, *Vehicle System Dynamics* 20(2): 57–78.

network, *Neurocomput.* 69(4-6): 449–465.

*and Computer Engineering Department* .

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learning, *IEEE Trans. Signal Process* 43(6): 1485–1497.

*System Dynamics* 24: 1–33.

38(3): 217–222.

30(1): 17–35.

21(10): 1599–1609.

7(1): 41–47.

*Netw.* 8(2): 227–236.

3(6): 889–898.

White-Smoke (2011).

Bridges are the crucial components of highway networks. In recent years, there has been growing awareness about the problems associated with the existing old bridges. Many of the existing bridges in service today were designed for less traffic, smaller vehicles, slower speeds and lighter traffic. Hence, they have become inadequate according to the current loading standards/codes of practice for design of highway bridges. Even in the case of newer bridges, deterioration caused by unforeseen service condition, adverse environmental actions and inadequate maintenance is causing great concern to bridge engineers. Bridge authority has the responsibility to maintain its bridges in a safe condition. To ensure safe and durable service, it is usual to perform periodic in-situ inspections. These inspections involve visual observations, non-destructive testing and partial destructive testing. The data collected from site and processed in laboratory, would be used to decide about suitable repair, strengthening or demolition of existing bridges. It is also evident that engineers and decision makers have to deal with large number of deficient bridges in years to come and it will be extremely demanding to decide the most deserving one to allot fund for timely retrofitting. Further, it is necessary to formulate a systematic method to assess the present and future needs of the existing bridges which would help the decision makers in identifying the most deserving bridges for improvement during a given period.

In view of this, several countries have initiated development of bridge management systems for assisting their decision makers in finding optimal strategies for maintenance, rehabilitation and replacement of bridges. Furthermore, it also has to ensure value for money by carrying out preventive work at appropriate time so that future maintenance needs are also kept at a minimum level. In a broader sense, the funding body has to consider the justification and priority for money to be spent on a multitude of expenditure areas. Decision makers and/or society at large should be able to choose whether to spend money on rehabilitating a bridge or to demolish it. The bridge engineers and the policy makers are being increasingly pressed to justify the funding order proposed to maintain the bridges. It shows the importance of an exclusive bridge management system. Bridge management is a rational and systematic approach for organising and carrying out the activities related to planning, design, construction, maintenance, rehabilitation and replacement of bridges.

Condition Ranking and Rating of Bridges Using Fuzzy Logic 177

the decision making methods, the Weighted Sum Model (WSM) is probably the best known and most widely used method of decision making, especially in single dimensional problem. If there are M alternatives and N criteria in a decision making problem, then the best alternative, A\*, is the one which satisfies (in the maximisation case) the following

> ij j j 1 a W

where, aij is the measure of performance of the ith alternative in terms of the jth decision criterion, and Wj is the weight of importance of the jth criterion. Further, Weighted Product Model (WPM) is very similar to WSM. The main difference is that it uses multiplication, instead of addition, to rank alternatives. Each alternative is compared with the others by multiplying a number of ratios, one for each criterion. Each ratio is raised to the power of the related weight of the corresponding criterion. Generally, in order to compare the two alternatives AK and AL, the following formula (Bridgman, 1922; Miller and Starr, 1969; Chen

*N*

where, N is number of criteria, aij is actual value (performance) of ith alternative in terms of jth criterion and Wj is weight of importance of the jth criterion. The analytic hierarchy process (AHP) was developed by Saaty (1980), based on an axiomatic foundation that has established its mathematical viability (Harker and Vargas 1990; Saaty, 1994). The diverse applications of the technique are due to its simplicity and ability to cope with complex decision making problems. The AHP methodology has been widely used for solving problems where definite quantitative measures are not available to support correct decisions. Zahedi (1986) provided an exhaustive survey of AHP methodology and its applications. The AHP attracted the interest of many researchers for long because of its easy applicability and interesting mathematical properties. In this chapter also, AHP, the wellproven technique, is used as a decision making tool because of its inherent strength in

The AHP deals with the construction of an M × N matrix (where M is the number of alternatives and N is the number of criteria) using the relative importance (weights) of the alternatives in terms of each criterion. The vector Xi =(ai1, ai2, ai3,….,aiN) for the ith alternative (i=1,2,3,…,M) is the eigenvector of an N × N reciprocal matrix which is determined through a sequence of pair-wise comparisons. Also, the elements in such a vector add-up to one. The AHP uses relative values instead of actual ones. Therefore, the AHP can be used in singleand multi-dimensional decision making problems. The analytic hierarchy model (AHM) begins with representing a complex problem as a hierarchy. At the top level of the hierarchy, the goal (objective) upon which the best decision should be made is placed. The next level of the hierarchy contains attributes or criteria that contribute to the quality of the

*j L* 

Wj Kj 1 j a a

for i= 1,2,….,M (1)

(2)

N

L A A =

**2.1 Formation of Analytic Hierarchy Model (AHM) for AHP** 

WSM = max *i*

expression (Fishburn, 1967)

A\*

and Hwang, 1992) can be used.

tackling complex problems.

R <sup>K</sup>

To decide upon all these matters, a systematic and logical way for prioritization of the bridges under consideration and rating of the most deserved one is needed (as shown in Fig. 1). The bridge condition rating is the datum for any bridge management system. The usefulness of a bridge management system and the accuracy of bridge rating rely upon the bridge condition data which constitute subjective judgment and intuition of the bridge inspector. So, a procedure like fuzzy logic would be useful to handle the uncertainty, imprecision and subjective judgment.

Fig. 1. Schematic representation of condition assessment of bridges

Before proceeding further, it is important to know the decision making tools useful for this type of problem. To provide the ready reference to the readers, few of the mostly used and appropriate models are discussed below.

#### **2. Different decision making methods**

One of the most crucial problems in many decision making methods is the precise evaluation of data. Very often, in real-life decision making applications, data are imprecise and fuzzy [Ben-Arieh and Triantaphyllou (1992), Tseng and Klein (1992)]. A decision maker may encounter difficulty in quantifying and processing linguistic statements. Therefore, it is desirable to develop decision making methods which can handle fuzzy data. It is equally important to evaluate the performance of the following decision making methods. Among

To decide upon all these matters, a systematic and logical way for prioritization of the bridges under consideration and rating of the most deserved one is needed (as shown in Fig. 1). The bridge condition rating is the datum for any bridge management system. The usefulness of a bridge management system and the accuracy of bridge rating rely upon the bridge condition data which constitute subjective judgment and intuition of the bridge inspector. So, a procedure like fuzzy logic would be useful to handle the uncertainty,

*Fuzzy Based Decision Support Tool* 

**Priority Ranking of** *n* **no. of bridges (Br1,Br2, . . . , Brn)** 

*Network Level*

*No*

**Different types of assessments**

**Bridges in consideration** 

**If major distresses observed** 

*Object Level*

**Detailed Inspections (Destructive, non-destructive and Chemical tests)** 

*Yes*

**Evaluation of inspection results**

**Assessment of condition** 

**Condition rating**

**Final result on condition evaluation of bridges** 

Before proceeding further, it is important to know the decision making tools useful for this type of problem. To provide the ready reference to the readers, few of the mostly used and

One of the most crucial problems in many decision making methods is the precise evaluation of data. Very often, in real-life decision making applications, data are imprecise and fuzzy [Ben-Arieh and Triantaphyllou (1992), Tseng and Klein (1992)]. A decision maker may encounter difficulty in quantifying and processing linguistic statements. Therefore, it is desirable to develop decision making methods which can handle fuzzy data. It is equally important to evaluate the performance of the following decision making methods. Among

Fig. 1. Schematic representation of condition assessment of bridges

imprecision and subjective judgment.

*Condition Assessment*

appropriate models are discussed below.

**2. Different decision making methods** 

the decision making methods, the Weighted Sum Model (WSM) is probably the best known and most widely used method of decision making, especially in single dimensional problem. If there are M alternatives and N criteria in a decision making problem, then the best alternative, A\*, is the one which satisfies (in the maximisation case) the following expression (Fishburn, 1967)

$$\mathbf{A}^\*\_{\rm WSM} = \max\_i \sum\_{\mathbf{i}=1}^N \mathbf{a}\_{\mathbf{i}\mathbf{j}} \mathbf{W}\_{\mathbf{j}} \quad \text{for } \mathbf{i} = 1, 2, ..., \mathbf{M} \tag{1}$$

where, aij is the measure of performance of the ith alternative in terms of the jth decision criterion, and Wj is the weight of importance of the jth criterion. Further, Weighted Product Model (WPM) is very similar to WSM. The main difference is that it uses multiplication, instead of addition, to rank alternatives. Each alternative is compared with the others by multiplying a number of ratios, one for each criterion. Each ratio is raised to the power of the related weight of the corresponding criterion. Generally, in order to compare the two alternatives AK and AL, the following formula (Bridgman, 1922; Miller and Starr, 1969; Chen and Hwang, 1992) can be used.

$$\mathbf{R}\left(\frac{\mathbf{A}\_{\mathbf{K}}}{\mathbf{A}\_{\mathbf{L}}}\right) = \prod\_{j=1}^{N} \left(\frac{\mathbf{a}\_{\mathbf{K}j}}{\mathbf{a}\_{Lj}}\right)^{\mathsf{M}\_{j}}\tag{2}$$

where, N is number of criteria, aij is actual value (performance) of ith alternative in terms of jth criterion and Wj is weight of importance of the jth criterion. The analytic hierarchy process (AHP) was developed by Saaty (1980), based on an axiomatic foundation that has established its mathematical viability (Harker and Vargas 1990; Saaty, 1994). The diverse applications of the technique are due to its simplicity and ability to cope with complex decision making problems. The AHP methodology has been widely used for solving problems where definite quantitative measures are not available to support correct decisions. Zahedi (1986) provided an exhaustive survey of AHP methodology and its applications. The AHP attracted the interest of many researchers for long because of its easy applicability and interesting mathematical properties. In this chapter also, AHP, the wellproven technique, is used as a decision making tool because of its inherent strength in tackling complex problems.

#### **2.1 Formation of Analytic Hierarchy Model (AHM) for AHP**

The AHP deals with the construction of an M × N matrix (where M is the number of alternatives and N is the number of criteria) using the relative importance (weights) of the alternatives in terms of each criterion. The vector Xi =(ai1, ai2, ai3,….,aiN) for the ith alternative (i=1,2,3,…,M) is the eigenvector of an N × N reciprocal matrix which is determined through a sequence of pair-wise comparisons. Also, the elements in such a vector add-up to one. The AHP uses relative values instead of actual ones. Therefore, the AHP can be used in singleand multi-dimensional decision making problems. The analytic hierarchy model (AHM) begins with representing a complex problem as a hierarchy. At the top level of the hierarchy, the goal (objective) upon which the best decision should be made is placed. The next level of the hierarchy contains attributes or criteria that contribute to the quality of the

Condition Ranking and Rating of Bridges Using Fuzzy Logic 179

analyst. Let w1, w2, w3, …………, wp be the real membership values of a fuzzy set with p members. Comparing objective k with objective l, the ratios kl can be assigned, and the RCP

The entry kl in RCP matrix represents the exact (and thus unknown) value of the comparison when the kth member is compared with the lth member. Each element k*<sup>l</sup>* (k*<sup>l</sup>* ) in the CDP matrix can be determined and the matrix will be formed such that

> 

1 1 *kl kl kl kl*

The Analytic Hierarchy Process (AHP) is mainly applied to the decision making problem with multiple evaluation criteria and uncertainty conditions. After hierarchical decomposition from different layers and through the quantitative judgment, the AHP is thus made a synthetic evaluation to reduce risk of wrong decision making. The AHP uses eigenvalue method to find the weights of different items. The eigen equation is adopted to construct the comparison matrix (Yu and Cheng, 1994, Liang et al., 2001) for finding the relative importance (weights) and orders of multiple objectives to an objective and the concept has already been successfully used to solve different types of decision making problems. The methodology involves the

A decision-maker provides the upper triangle of the comparison matrix (as shown in Table 1), while reciprocals are placed in the lower triangle which do not need any further judgment. The diagonal elements of the matrix are always equal to one. Assuming that any item group consists of A1, A2, A3, ….An items, the comparison matrix is constructed and then relative weights of items (A*ij*) of the group are evaluated by comparing objective *i* with objective *j*, the ratios kl can be assigned, and the real continuous pair-wise (RCP) matrix of order *p p* is constructed. It can be proved that consistent reciprocal matrix '[A]' has rank 1

The same equation also states that in the perfectly consistent case (i.e. A*ij* = A*ik* A*kj*), the vector w, with the membership values of the elements 1,2,3,….,n is the principal right-

In most of the real world problems, the pair-wise comparisons are not perfect, that is, the entry A*ij* might deviate from the ratio of the real membership values W*i* and W*j* (i.e. W*i* /

[A]w = nw Where, w is an eigenvector (5)

l w w 

k,l = 1,p (3)

(4)

RCP = A pp = [kl] = k

is minimum. Any other norm may also be assumed as

**3. Condition evaluation of existing bridges through prioritization** 

matrix (p p) is constructed as

( ) *kl kl* 

following operations.

**3.1 Relative importance (weights) of items** 

with non-zero eigenvalue () = n. Then, we have

eigenvector (after normalisation) of matrix [A].

**3.2 Check for consistency of comparison matrices** 

decisions. Each attribute may be decomposed into more detailed attributes (indices). After the hierarchical network is constructed, one can determine the weights (importance measures) of the elements at each level of the decision hierarchy, and synthesize the weights to determine the relative importance (weights) of decision alternatives. First, a comparison matrix, which includes first (lowest) level elements of the hierarchy, is constructed. Then, a ratio scale through pair-wise comparison of each pair of criteria with respect to the overall goal is performed. The relative importance (weight) of each criterion is estimated using an eigenvector approach or other methods. Then, the relative importance (weight) of each alternative with respect to each criterion is determined using similar pair-wise comparisons. Here, it is important to note that the efficiency of AHP greatly depends on the accuracy with which pair-wise weights of items are assigned during the formation of comparison matrix. For pair-wise assignment of weights for items, there is a need for a scale for relative quantification of items.

#### **2.2 Scales for quantifying pair-wise comparisons**

One of the most vital and crucial steps in decision-making methods is the accurate estimation of the pertinent data. Very often, these data are not known in terms of absolute values. Therefore, many decision-making methods attempt to determine the relative importance (weight) of each alternative involved in a given decision-making problem. Consider the case of having a single decision criterion and a set of N alternatives denoted as Ai (i = 1, 2, 3,…, N). The decision maker wants to determine the relative performance of the alternatives under each criterion. Here, one may consider the N alternatives as the members of a fuzzy set. Then, the degree of membership of element (i.e. alternative) Ai expresses the degree to which alternative Ai meets the criterion. This is also the approach considered by Federov et al. (1982) and Chen and Hwang (1992) and was also discussed by Saaty (1994). All the methods which use the pair-wise comparison approach eventually express the qualitative answers of a decision maker as some numbers. Pair-wise comparisons are quantified by using a scale. Such a scale is one-to-one mapping between the set of discrete linguistic choices available to the decision maker and a discrete set of numbers which represent the importance or weight of the previous linguistic choices. There are two major approaches in developing such scales. The first approach is based on the linear scale and the other is based on exponential scale [Roberts (1979), Lootsma (1991)]. It is easier to use linear scale to translate the weight of an item/element over the other. Therefore, in this study, the linear scale has been used to assign the importance/weight of items or elements under each decision layer.

#### **2.3 Real Continuous Pair-wise (RCP) and Closest Discrete Pair-wise (CDP) matrices**

A procedure is required for obtaining comparison matrix from the relative importance (weights) for a group of elements, using a suitable scale, based on pair-wise comparisons. It involves the formulation of real continuous pair-wise (RCP) and the closest discrete pairwise (CDP) matrices. Reciprocal matrices with pair-wise comparisons were used for extracting all the pertinent information from a decision maker. Each entry in these matrices represents numerically the value of a pair-wise comparison between two alternatives with respect to a single criterion. For a problem that has 'p' objectives, a scale is constructed for rating these objectives as to their importance with respect to the decision as seen by the

decisions. Each attribute may be decomposed into more detailed attributes (indices). After the hierarchical network is constructed, one can determine the weights (importance measures) of the elements at each level of the decision hierarchy, and synthesize the weights to determine the relative importance (weights) of decision alternatives. First, a comparison matrix, which includes first (lowest) level elements of the hierarchy, is constructed. Then, a ratio scale through pair-wise comparison of each pair of criteria with respect to the overall goal is performed. The relative importance (weight) of each criterion is estimated using an eigenvector approach or other methods. Then, the relative importance (weight) of each alternative with respect to each criterion is determined using similar pair-wise comparisons. Here, it is important to note that the efficiency of AHP greatly depends on the accuracy with which pair-wise weights of items are assigned during the formation of comparison matrix. For pair-wise assignment of weights for items, there is a need for a scale for relative

One of the most vital and crucial steps in decision-making methods is the accurate estimation of the pertinent data. Very often, these data are not known in terms of absolute values. Therefore, many decision-making methods attempt to determine the relative importance (weight) of each alternative involved in a given decision-making problem. Consider the case of having a single decision criterion and a set of N alternatives denoted as Ai (i = 1, 2, 3,…, N). The decision maker wants to determine the relative performance of the alternatives under each criterion. Here, one may consider the N alternatives as the members of a fuzzy set. Then, the degree of membership of element (i.e. alternative) Ai expresses the degree to which alternative Ai meets the criterion. This is also the approach considered by Federov et al. (1982) and Chen and Hwang (1992) and was also discussed by Saaty (1994). All the methods which use the pair-wise comparison approach eventually express the qualitative answers of a decision maker as some numbers. Pair-wise comparisons are quantified by using a scale. Such a scale is one-to-one mapping between the set of discrete linguistic choices available to the decision maker and a discrete set of numbers which represent the importance or weight of the previous linguistic choices. There are two major approaches in developing such scales. The first approach is based on the linear scale and the other is based on exponential scale [Roberts (1979), Lootsma (1991)]. It is easier to use linear scale to translate the weight of an item/element over the other. Therefore, in this study, the linear scale has been used to assign the importance/weight of items or elements under each

**2.3 Real Continuous Pair-wise (RCP) and Closest Discrete Pair-wise (CDP) matrices**  A procedure is required for obtaining comparison matrix from the relative importance (weights) for a group of elements, using a suitable scale, based on pair-wise comparisons. It involves the formulation of real continuous pair-wise (RCP) and the closest discrete pairwise (CDP) matrices. Reciprocal matrices with pair-wise comparisons were used for extracting all the pertinent information from a decision maker. Each entry in these matrices represents numerically the value of a pair-wise comparison between two alternatives with respect to a single criterion. For a problem that has 'p' objectives, a scale is constructed for rating these objectives as to their importance with respect to the decision as seen by the

quantification of items.

decision layer.

**2.2 Scales for quantifying pair-wise comparisons** 

analyst. Let w1, w2, w3, …………, wp be the real membership values of a fuzzy set with p members. Comparing objective k with objective l, the ratios kl can be assigned, and the RCP matrix (p p) is constructed as

$$\text{RCP} = \text{A}\_{\text{p} \times \text{p}} = \begin{bmatrix} \alpha\_{\text{kl}} \end{bmatrix} = \begin{bmatrix} \mathbf{w}\_{\text{k}} \\ \hline \mathbf{w}\_{\text{l}} \end{bmatrix} \quad \text{k}, \text{l} = \text{1}, \text{p} \tag{3}$$

The entry kl in RCP matrix represents the exact (and thus unknown) value of the comparison when the kth member is compared with the lth member. Each element k*<sup>l</sup>* (k*<sup>l</sup>* ) in the CDP matrix can be determined and the matrix will be formed such that ( ) *kl kl* is minimum. Any other norm may also be assumed as

$$\left| \frac{\alpha\_{kl}}{1 + \alpha\_{kl}} - \frac{\beta\_{kl}}{1 + \beta\_{kl}} \right| \tag{4}$$

#### **3. Condition evaluation of existing bridges through prioritization**

The Analytic Hierarchy Process (AHP) is mainly applied to the decision making problem with multiple evaluation criteria and uncertainty conditions. After hierarchical decomposition from different layers and through the quantitative judgment, the AHP is thus made a synthetic evaluation to reduce risk of wrong decision making. The AHP uses eigenvalue method to find the weights of different items. The eigen equation is adopted to construct the comparison matrix (Yu and Cheng, 1994, Liang et al., 2001) for finding the relative importance (weights) and orders of multiple objectives to an objective and the concept has already been successfully used to solve different types of decision making problems. The methodology involves the following operations.

#### **3.1 Relative importance (weights) of items**

A decision-maker provides the upper triangle of the comparison matrix (as shown in Table 1), while reciprocals are placed in the lower triangle which do not need any further judgment. The diagonal elements of the matrix are always equal to one. Assuming that any item group consists of A1, A2, A3, ….An items, the comparison matrix is constructed and then relative weights of items (A*ij*) of the group are evaluated by comparing objective *i* with objective *j*, the ratios kl can be assigned, and the real continuous pair-wise (RCP) matrix of order *p p* is constructed. It can be proved that consistent reciprocal matrix '[A]' has rank 1 with non-zero eigenvalue () = n. Then, we have

$$\text{[A]}\mathbf{w} = \mathbf{n}\mathbf{w}\text{ Where, }\mathbf{w}\text{ is an eigenvector}\tag{5}$$

The same equation also states that in the perfectly consistent case (i.e. A*ij* = A*ik* A*kj*), the vector w, with the membership values of the elements 1,2,3,….,n is the principal righteigenvector (after normalisation) of matrix [A].

#### **3.2 Check for consistency of comparison matrices**

In most of the real world problems, the pair-wise comparisons are not perfect, that is, the entry A*ij* might deviate from the ratio of the real membership values W*i* and W*j* (i.e. W*i* /

Condition Ranking and Rating of Bridges Using Fuzzy Logic 181

*x*

where, *l ≤ m ≤ u* , and *l* and *u* stand for the lower and upper values of the support for the decision of the fuzzy number M, respectively, and m for the modal value. In this study, the basic mathematical operations on fuzzy triangular numbers developed by Laarhoven and Pedrycz (1983) are followed. In decision problems, the maximum and minimum membership function suggested by Zadeh (1973) are adopted and expressed in the

1 *f*(x) ≤ inf(*f*)

1 *f*(x) ≥ sup(*f*)

where sup(*f*) and inf(*f*) are the superior and inferior values of *f*(x) respectively. It is understandable that Eq. (9) is a membership function with monotonic decrease whereas Eq. (10) is a membership function with monotonic increase. The significance of Eq. (9) is that the less the value is, more requirement for repair whereas the meaning of Eq. (10) is just the opposite of Eq. (9). An evaluation method can be developed by separating bridge deterioration into D (degree), E (extent), R (relevance) and U (urgency) for assessment. A combination of visual inspection, field and laboratory testing may be employed for determining the item estimation indices of bridges considered for condition assessment. Based on inspection results of all the items, the condition index (CoI) is calculated by using

CoI = ( ) *i i*

Where, *wi* is the weight of each bridge item and is greater than 1, and Ici is calculated as

*Ici* =

item condition estimation index for each item and is calculated as

*i Ic w w* 

> *ii Ic n*

in which n is the number of relevant inspection items for a particular bridge, and *Icii* is the

 *Icii* = *a a DERU* (13)

*ml ml <sup>x</sup>* [*l, m*]

0 otherwise

*mu mu <sup>x</sup>* [*m, u*] (8)

inf(*f*) *<sup>f</sup>*(x) sup(*f*) (9)

inf(*f*) < *f*(x) < sup(*f*) (10)

(11)

(12)

 <sup>1</sup> *<sup>l</sup> x*

m (x) = <sup>1</sup> *<sup>l</sup>*

following form.

and

(x)= sup( ) ( )

(x)= ( ) inf( )

sup( ) inf( ) *f f x f f* 

sup( ) inf( ) *f x f f f* 

0 *f*(x) ≥ sup(*f*)

0 *f*(x) ≤ inf(*f*)

W*j*). In a non-consistent case, the expression A*ij* = A*ik* × A*kj* does not hold good for all the possible combinations. Now, the new matrix [A] can be considered as a perturbation of the previous consistent case when the entries A*ij* change slightly, then the eigenvalues change in the similar fashion (Saaty, 1994). Moreover, the maximum eigenvalue is close to n (greater than n), while the remaining eigenvalues are close to zero. Thus, in order to find the membership values in the non-consistent cases, one should find an eigenvector that corresponds to the largest eigenvalue max. That is to say, one must find the principal righteigenvector W that satisfies

$$\mathbf{A}\mathbf{W} = \lambda\_{\text{max}} \mathbf{W} \qquad\quad\text{where }\lambda\_{\text{max}} \approx \mathbf{n} \tag{6}$$

The consistency ratio (CR) is obtained by first estimating max of matrix [A] Then, Saaty (1994) defined the consistency index (CI) of the matrix '[A]' as

$$\mathbf{CI} = (\mathbf{\bar{\lambda}}\_{\text{-max}} \mathbf{\bar{\star}} \mathbf{n}) / (\mathbf{\bar{\iota}} \mathbf{\bar{\iota}} \mathbf{1}) \tag{7}$$

Then, the consistency ratio (CR) is obtained by dividing CI with the random consistency index (RCI) as shown in Table 2 (proposed by Saaty, 1994). Each RCI is an average random consistency index derived from a sample of size 500 of randomly generated reciprocal matrices. If the previous approach yields a CR greater than 0.10 then a re-examination of the pair-wise judgments is recommended until a CR less than or equal to 0.10 is achieved.


Table 1. Comparison Matrix


Table 2. RCI values of sets of different order 'n'

#### **3.3 Fuzzy synthetic evaluation of estimation indices for items**

Most of the decision making in the real world takes place in a situation in which the pertinent data and the sequence of possible actions are not precisely known. Therefore, it is very important to adopt fuzzy data to express such situations in decision making problems. In order to fuzzify the crisp decision making methods, it is important to know how fuzzy operations are used on fuzzy numbers. Fuzzy operation in decision making was first introduced by Dubois and Prade (1979) and Boender et al. (1989) presented a fuzzy version of the AHP. For fuzzy numbers, triangular fuzzy numbers (that is, fuzzy numbers with lower, modal and upper values) are preferred, because they are simpler than trapezoidal fuzzy numbers. A fuzzy number M on R (-, +) is defined by Dubois and Prade, 1979 to be a fuzzy triangular number if its membership function m: R [0,1] is equal to

$$\mu\_{\text{m}}\left(\mathbf{x}\right) = \begin{cases} \frac{1}{m-l}\mathbf{x} - \frac{l}{m-l} & \mathbf{x} \in [l, m] \\ \frac{1}{m-u}\mathbf{x} - \frac{l}{m-u} & \mathbf{x} \in [m, u] \\ 0 & \text{otherwise} \end{cases} \tag{8}$$

where, *l ≤ m ≤ u* , and *l* and *u* stand for the lower and upper values of the support for the decision of the fuzzy number M, respectively, and m for the modal value. In this study, the basic mathematical operations on fuzzy triangular numbers developed by Laarhoven and Pedrycz (1983) are followed. In decision problems, the maximum and minimum membership function suggested by Zadeh (1973) are adopted and expressed in the following form.

$$\mu(\mathbf{x}) = \begin{cases} 1 & f(\mathbf{x}) \le \inf \{ f \} \\ \frac{\sup(f) - f(\mathbf{x})}{\sup(f) - \inf(f)} & \inf(f) < f(\mathbf{x}) < \sup(f) \\ 0 & f(\mathbf{x}) \ge \sup(f) \end{cases} \tag{9}$$

and

180 Fuzzy Logic – Emerging Technologies and Applications

W*j*). In a non-consistent case, the expression A*ij* = A*ik* × A*kj* does not hold good for all the possible combinations. Now, the new matrix [A] can be considered as a perturbation of the previous consistent case when the entries A*ij* change slightly, then the eigenvalues change in the similar fashion (Saaty, 1994). Moreover, the maximum eigenvalue is close to n (greater than n), while the remaining eigenvalues are close to zero. Thus, in order to find the membership values in the non-consistent cases, one should find an eigenvector that corresponds to the largest eigenvalue max. That is to say, one must find the principal right-

The consistency ratio (CR) is obtained by first estimating max of matrix [A] Then, Saaty

 CI=(max-n)/(n-1) (7) Then, the consistency ratio (CR) is obtained by dividing CI with the random consistency index (RCI) as shown in Table 2 (proposed by Saaty, 1994). Each RCI is an average random consistency index derived from a sample of size 500 of randomly generated reciprocal matrices. If the previous approach yields a CR greater than 0.10 then a re-examination of the pair-wise judgments is recommended until a CR less than or equal to 0.10 is achieved.

> B A1 A2 A3 …. An A1 A11 A12 A13 …. A1n A2 A21 A22 A23 …. A2n A3 A31 A32 A33 …. A3n …. …. …. …. …. …. An An1 An2 An3 …. Ann

n 1 2 3 4 5 6 7 8 9 ≥10 RCI 0 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.56

Most of the decision making in the real world takes place in a situation in which the pertinent data and the sequence of possible actions are not precisely known. Therefore, it is very important to adopt fuzzy data to express such situations in decision making problems. In order to fuzzify the crisp decision making methods, it is important to know how fuzzy operations are used on fuzzy numbers. Fuzzy operation in decision making was first introduced by Dubois and Prade (1979) and Boender et al. (1989) presented a fuzzy version of the AHP. For fuzzy numbers, triangular fuzzy numbers (that is, fuzzy numbers with lower, modal and upper values) are preferred, because they are simpler than trapezoidal fuzzy numbers. A fuzzy number M on R (-, +) is defined by Dubois and Prade, 1979 to

be a fuzzy triangular number if its membership function m: R [0,1] is equal to

(1994) defined the consistency index (CI) of the matrix '[A]' as

AW = max W where max n (6)

eigenvector W that satisfies

Table 1. Comparison Matrix

Table 2. RCI values of sets of different order 'n'

**3.3 Fuzzy synthetic evaluation of estimation indices for items** 

$$\mu(\mathbf{x}) = \begin{cases} 1 & f(\mathbf{x}) \ge \sup(f) \\ \frac{f(\mathbf{x}) - \inf(f)}{\sup(f) - \inf(f)} & \inf(f) < f(\mathbf{x}) < \sup(f) \\ 0 & f(\mathbf{x}) \le \inf(f) \end{cases} \tag{10}$$

where sup(*f*) and inf(*f*) are the superior and inferior values of *f*(x) respectively. It is understandable that Eq. (9) is a membership function with monotonic decrease whereas Eq. (10) is a membership function with monotonic increase. The significance of Eq. (9) is that the less the value is, more requirement for repair whereas the meaning of Eq. (10) is just the opposite of Eq. (9). An evaluation method can be developed by separating bridge deterioration into D (degree), E (extent), R (relevance) and U (urgency) for assessment. A combination of visual inspection, field and laboratory testing may be employed for determining the item estimation indices of bridges considered for condition assessment. Based on inspection results of all the items, the condition index (CoI) is calculated by using

$$\text{CoI} = \frac{\sum \text{(lc}\_i \times w\_i)}{\sum w\_i} \tag{11}$$

Where, *wi* is the weight of each bridge item and is greater than 1, and Ici is calculated as

$$Ic\_i = \frac{\sum Ic\_{ii}}{n} \tag{12}$$

in which n is the number of relevant inspection items for a particular bridge, and *Icii* is the item condition estimation index for each item and is calculated as

$$I\_{\mathbb{C}\bar{u}} = D \times E \times \mathbb{R}^d \times \mathbb{U}^d \tag{13}$$

Condition Ranking and Rating of Bridges Using Fuzzy Logic 183

bridges. Zhao and Chen (2002) developed a fuzzy rule-based inference system for bridge damage diagnosis and prediction which aims at providing bridge designers with valuable information about the impact of design factors on bridge deterioration. Kawamura and Miyamoto (2003) presented a new approach for developing a concrete bridge rating expert system for deteriorated concrete bridges, using multi-layer neural networks. To evaluate the condition of different structures using fuzzy logic, the proposed methods are either too simplistic [Qian (1992); Liang et al. (2001)] or very complex [Jwu et al. (1991); Kawamura and Miyamoto (2003)]. In this chapter, a systematic procedure and formulations have been proposed for condition rating of existing bridges using fuzzy mathematics combined with eigenvector based priority setting technique. From the review of literature, authors felt that the existing methodologies for condition rating of existing bridges may be difficult to follow for a practical application. In view of this, in this chapter, a methodology for condition rating of bridges is described in steps that can easily be followed for practical applications. The methodology and its application are demonstrated through a case study and the details

Some important issues and the methodology for the development of a systematic, fast and reliable evaluation system for rating of existing reinforced concrete bridges have been

Towards a systematic rating of existing bridges, the essential requirement is the input data from the bridge inspector that consist of the ratings and importance factors for the relevant

Bridge inspection involves the use of various evaluation techniques in order to assess the physical condition of bridges. The Bridge Inspector's Training manual 90 (FHWA 1991), published by the US Department of Transportation, provides the basic guidelines for bridge inspection. Bridge components and their constituent elements, different types of bridge deterioration and the common causes are discussed in this manual. It also provides procedures for rating the condition of various elements. In this study also, as specified in Bridge Inspector's Training manual, bridge is divided into three major components, namely, 'deck', 'superstructure' and 'substructure'. Each component is further divided into number of elements. The deck, superstructure and substructure have 13, 16 and 20 elements respectively as shown in Table 3. The bridge inspector is required to assess the condition of each element individually. The rating evaluation for that particular component is carried out based on the rating of the constituent elements. This process is repeated for all the three components towards final rating of the bridge. To a large extent, rating of the elements is based on the experience, intuition and personal judgment of the inspector. Nevertheless, although the condition assessment of each element requires the inspector's personal judgment, general guidelines on how to assess the condition of the various elements are described in the

elements of bridge which would reflect the overall condition of a bridge as a whole.

are presented in the chapter.

**4.1.1 Inspection data** 

illustrated in the following sections.

**4.1 Unified approach for condition rating of existing bridge** 

**4.1.2 Inspector's observation and rating of elements** 

#### **3.4 Condition ranking of existing bridges**

If Rn,m (in which n= 1,2….no of criteria layers, and m=1,2,…of items in each index layer) denotes the membership degrees of estimation indices of the items under index layer and *Wn* stands for the relative weights of items under index layer (calculated by using Eq. 3), then the relationship between Rn,m and *Wn* can be presented by

$$
\overline{D}\_n = \overline{\mathcal{W}}\_n \mathcal{R}\_{n.m} \tag{14}
$$

where the value *Dn* in Eq. 14 is the fuzzy synthesis evaluation matrix. The purpose of *Dn* is to construct the membership function for each alternative of evaluation set. The membership degrees of estimation indices, Rn.m, can be formulated based upon the decision makers choice in using the pessimistic- or optimistic- functions as stated in Eqs. 9 and 10, respectively.

Based on the fuzzy mathematics theory, the fuzzy synthesis evaluation result, *B* , of any factor can be expressed as

$$
\overline{B} = \overline{A}\_n.\overline{D}\_n\tag{15}
$$

where, *An* is the weight vector. The prioritization or optimum repair order can be determined by using Eq. (15). The more the value of *B* has, the better the priority selection to decision making objective is.

#### **4. Application of fuzzy logic for condition rating of bridges**

The aim of the bridge condition rating is to evaluate the structural strength and serviceability condition of an existing bridge. Fuzzy set theory was specially defined to analyse the linguistic data within the formal mathematical framework. After the publication on fuzzy sets by Zadeh (1965), fuzzy mathematics was used extensively for numerous applications. Brown and Yao (1983) described the methodology of application of fuzzy sets in structural engineering. Tee et al. (1988) suggested a fuzzy mathematical approach for evaluation of bridges. Shiaw and Huang (1990) adopted the limit state design principle combined with fuzzy evaluation and random variable analysis to determine the bearing capacity index and degree of safety for bridges. Jwu et al. (1991) used fuzzy mathematics to determine the reliability of a wharf structure. In order to enhance the evaluation performance, the grade partition method was suggested. Tee and Bowman (1991) presented bridge condition assessment model that is based on resolution identity of fuzzy sets. Qian (1992) used the concept of fuzzy sets to evaluate the damage grade of existing bridges. Yu and Cheng (1994) presented a fuzzy based interactive comparison matrix approach for making group decision with multiple objectives. Wang (1996) provided a multi-target and multi-person evaluation method for structural durability. Melhem and Aturaliya (1996) proposed a model for condition rating of bridges using an eigenvector based priority setting. Liang et al. (2001) used fuzzy mathematics to build a damage evaluation methodology for existing reinforced concrete bridges. Liang et al. (2002) proposed grey and regression models for predicting the remaining service life of existing reinforced concrete bridges. Zhao and Chen (2002) developed a fuzzy rule-based inference system for bridge damage diagnosis and prediction which aims at providing bridge designers with valuable information about the impact of design factors on bridge deterioration. Kawamura and Miyamoto (2003) presented a new approach for developing a concrete bridge rating expert system for deteriorated concrete bridges, using multi-layer neural networks. To evaluate the condition of different structures using fuzzy logic, the proposed methods are either too simplistic [Qian (1992); Liang et al. (2001)] or very complex [Jwu et al. (1991); Kawamura and Miyamoto (2003)]. In this chapter, a systematic procedure and formulations have been proposed for condition rating of existing bridges using fuzzy mathematics combined with eigenvector based priority setting technique. From the review of literature, authors felt that the existing methodologies for condition rating of existing bridges may be difficult to follow for a practical application. In view of this, in this chapter, a methodology for condition rating of bridges is described in steps that can easily be followed for practical applications. The methodology and its application are demonstrated through a case study and the details are presented in the chapter.

#### **4.1 Unified approach for condition rating of existing bridge**

Some important issues and the methodology for the development of a systematic, fast and reliable evaluation system for rating of existing reinforced concrete bridges have been illustrated in the following sections.

#### **4.1.1 Inspection data**

182 Fuzzy Logic – Emerging Technologies and Applications

If Rn,m (in which n= 1,2….no of criteria layers, and m=1,2,…of items in each index layer) denotes the membership degrees of estimation indices of the items under index layer and *Wn* stands for the relative weights of items under index layer (calculated by using Eq. 3),

where the value *Dn* in Eq. 14 is the fuzzy synthesis evaluation matrix. The purpose of *Dn* is to construct the membership function for each alternative of evaluation set. The membership degrees of estimation indices, Rn.m, can be formulated based upon the decision makers choice in using the pessimistic- or optimistic- functions as stated in Eqs. 9 and 10,

Based on the fuzzy mathematics theory, the fuzzy synthesis evaluation result, *B* , of any

where, *An* is the weight vector. The prioritization or optimum repair order can be determined by using Eq. (15). The more the value of *B* has, the better the priority selection

The aim of the bridge condition rating is to evaluate the structural strength and serviceability condition of an existing bridge. Fuzzy set theory was specially defined to analyse the linguistic data within the formal mathematical framework. After the publication on fuzzy sets by Zadeh (1965), fuzzy mathematics was used extensively for numerous applications. Brown and Yao (1983) described the methodology of application of fuzzy sets in structural engineering. Tee et al. (1988) suggested a fuzzy mathematical approach for evaluation of bridges. Shiaw and Huang (1990) adopted the limit state design principle combined with fuzzy evaluation and random variable analysis to determine the bearing capacity index and degree of safety for bridges. Jwu et al. (1991) used fuzzy mathematics to determine the reliability of a wharf structure. In order to enhance the evaluation performance, the grade partition method was suggested. Tee and Bowman (1991) presented bridge condition assessment model that is based on resolution identity of fuzzy sets. Qian (1992) used the concept of fuzzy sets to evaluate the damage grade of existing bridges. Yu and Cheng (1994) presented a fuzzy based interactive comparison matrix approach for making group decision with multiple objectives. Wang (1996) provided a multi-target and multi-person evaluation method for structural durability. Melhem and Aturaliya (1996) proposed a model for condition rating of bridges using an eigenvector based priority setting. Liang et al. (2001) used fuzzy mathematics to build a damage evaluation methodology for existing reinforced concrete bridges. Liang et al. (2002) proposed grey and regression models for predicting the remaining service life of existing reinforced concrete

**4. Application of fuzzy logic for condition rating of bridges** 

*D WR n n nm* . (14)

. *B AD n n* (15)

**3.4 Condition ranking of existing bridges** 

respectively.

factor can be expressed as

to decision making objective is.

then the relationship between Rn,m and *Wn* can be presented by

Towards a systematic rating of existing bridges, the essential requirement is the input data from the bridge inspector that consist of the ratings and importance factors for the relevant elements of bridge which would reflect the overall condition of a bridge as a whole.

#### **4.1.2 Inspector's observation and rating of elements**

Bridge inspection involves the use of various evaluation techniques in order to assess the physical condition of bridges. The Bridge Inspector's Training manual 90 (FHWA 1991), published by the US Department of Transportation, provides the basic guidelines for bridge inspection. Bridge components and their constituent elements, different types of bridge deterioration and the common causes are discussed in this manual. It also provides procedures for rating the condition of various elements. In this study also, as specified in Bridge Inspector's Training manual, bridge is divided into three major components, namely, 'deck', 'superstructure' and 'substructure'. Each component is further divided into number of elements. The deck, superstructure and substructure have 13, 16 and 20 elements respectively as shown in Table 3. The bridge inspector is required to assess the condition of each element individually. The rating evaluation for that particular component is carried out based on the rating of the constituent elements. This process is repeated for all the three components towards final rating of the bridge. To a large extent, rating of the elements is based on the experience, intuition and personal judgment of the inspector. Nevertheless, although the condition assessment of each element requires the inspector's personal judgment, general guidelines on how to assess the condition of the various elements are described in the

Condition Ranking and Rating of Bridges Using Fuzzy Logic 185

importance reported by Melhem and Aturaliya (1996) are used in the present study for structural importance factors of the elements under each component of the bridge. In this study, a scale of 1-9 has been considered for rating of the elements. An element with rating value of 9 signifies the best possible condition without distress and the descending rating numbers represent the increased degree of distress. The fuzzy membership values of structural importance for the elements of deck, superstructure and substructure are given in Tables 4, 5 and 6 respectively. From Tables 4 - 6, it may be noted that the mean value of the importance of an element increases as the physical condition deteriorates. For example, importance of deck concrete with rating 1 is 0.96, whereas its importance is 0.42 when the

If Rn is a fuzzy set, representing rating of an element (where 'n' represents rating number i.e.

 Rn = m(rm) rm (m = 0,1,2,….,9) (16) where, (r) is a membership function representing the degree of membership of any fuzzy set and 0 ≤ ≤ 1. The function as described in Eq. (16) quantifies the ambiguity associated with the rating of any element of a bridge. Any rating number can be represented using

**Item 0 1 2 3 4 5 6 7 8 9**  1 wearing coat 1 0.90 0.80 0.70 0.61 0.51 0.45 0.33 0.23 0.17

concrete 1 0.96 0.92 0.89 0.85 0.81 0.77 0.72 0.50 0.42

joint 1 0.92 0.85 0.77 0.70 0.62 0.55 0.47 0.38 0.30

Table 4. Mean values of the structural importance for the deck elements for different rating

3 curbs 1 0.85 0.70 0.55 0.40 0.25 0.20 0.14 0.10 0.08 4 median 1 0.85 0.70 0.54 0.39 0.24 0.21 0.14 0.11 0.09 5 sidewalks 1 0.88 0.76 0.64 0.52 0.40 0.33 0.25 0.17 0.14 6 parapets 1 0.88 0.76 0.63 0.51 0.39 0.33 0.26 0.19 0.19 7 railing 1 0.88 0.76 0.65 0.53 0.41 0.35 0.26 0.19 0.16 8 paint 1 0.87 0.74 0.61 0.48 0.35 0.31 0.24 0.18 0.15 9 drains 1 0.90 0.80 0.70 0.61 0.51 0.45 0.35 0.29 0.22 10 lighting 1 0.86 0.72 0.57 0.43 0.29 0.27 0.20 0.16 0.15 11 utilities 1 0.85 0.70 0.55 0.40 0.25 0.23 0.17 0.13 0.11 12 joint leakage 1 0.91 0.82 0.72 0.63 0.54 0.49 0.41 0.34 0.28

Mean values of Structural Importance

n =0,1,…..9), the general form of the membership function can be formed as follows:

**4.2 Fuzzification of input data obtained from bridge inspectors** 

fuzzy membership function (Emami et al. 1998).

**Rating**

rating is 9.

SL No.

<sup>2</sup>deck

<sup>13</sup>expansion


inspector's manual. Hence, while two competent bridge inspectors may differ on the rating of a given element, but their difference in the rating would not be significant.

Table 3. Decomposition of a bridge into elements with observed ratings (**R**)

#### **4.1.3 Evaluation of importance factors**

In a bridge condition evaluation, rating of each element under a particular component does not influence the component's overall structural condition rating in a similar degree. A well trained inspector or the concerned expert determines the structural importance of different elements of all components of a bridge. The importance factor of element is not constant but varies with the degree of distress sustained by the element under consideration. Hence, determination of structural importance factors for various bridge elements is not an easy task. The knowledge gained by the bridge inspectors and experts through many years of design and inspection experience can not be obtained directly through structural analysis, although analysis can provide general trends of the behaviour of damaged members.

So, the importance factors for the elements at various deterioration stages should be evolved from the response of competent bridge inspectors/experts. These membership functions for structural importance were originally constructed through a survey among a number of bridge engineers and inspectors (Tee et al., 1988). Then, the collected data was statistically processed and the mean was presented by Melhem and Aturaliya (1996). As membership functions for structural importance corresponding to different ratings of elements/components is not bridge specific, membership functions for structural

inspector's manual. Hence, while two competent bridge inspectors may differ on the rating

1. Wearing Surface 8 1. Bearing devices 5 1. Bridge seats 6 2. Deck condition 9 2. Stringers 2. Wings 5 3. Kerbs 6 3. Girders 4 3. Back wall 6 4. Median 9 4. Floor beams 7 4. Footings 5. Sidewalks 8 5. Trusses 5. Piles 7 6. Parapets 9 6. Paint 5 6. Erosion 8 7. Railings 6 7. Machinery 7. Settlements 9 8. Paint 7 8. Rivets-Bolts 8. Pier-cap 2 9. Drains 8 9. Welds 2 9. Pier-column 5 10. Lighting 9 10. Rust 4 10. Pier-footing 3 11. Utilities 8 11. Timber decay 11. Pier-piles 6 12. Joint leakage 5 12. Concrete cracks 5 12. Pier-scour 5 13. Expansion joints 9 13. Collision damage 6 13. Pier-settlement 6

17. Timber decay

In a bridge condition evaluation, rating of each element under a particular component does not influence the component's overall structural condition rating in a similar degree. A well trained inspector or the concerned expert determines the structural importance of different elements of all components of a bridge. The importance factor of element is not constant but varies with the degree of distress sustained by the element under consideration. Hence, determination of structural importance factors for various bridge elements is not an easy task. The knowledge gained by the bridge inspectors and experts through many years of design and inspection experience can not be obtained directly through structural analysis,

although analysis can provide general trends of the behaviour of damaged members.

So, the importance factors for the elements at various deterioration stages should be evolved from the response of competent bridge inspectors/experts. These membership functions for structural importance were originally constructed through a survey among a number of bridge engineers and inspectors (Tee et al., 1988). Then, the collected data was statistically processed and the mean was presented by Melhem and Aturaliya (1996). As membership functions for structural importance corresponding to different ratings of elements/components is not bridge specific, membership functions for structural

Table 3. Decomposition of a bridge into elements with observed ratings (**R**)

**4.1.3 Evaluation of importance factors** 

**Deck R Superstructure R Substructure R** 

 14. Deflection 5 14. Pier-bents 4 15. Member alignment 7 15. Concrete cracks 5 16. Vibrations 6 16. Steel corrosion 8

 18. Debris seats 5 19. Paint 6 20. Collision damage 5

Note: - not applicable

of a given element, but their difference in the rating would not be significant.

importance reported by Melhem and Aturaliya (1996) are used in the present study for structural importance factors of the elements under each component of the bridge. In this study, a scale of 1-9 has been considered for rating of the elements. An element with rating value of 9 signifies the best possible condition without distress and the descending rating numbers represent the increased degree of distress. The fuzzy membership values of structural importance for the elements of deck, superstructure and substructure are given in Tables 4, 5 and 6 respectively. From Tables 4 - 6, it may be noted that the mean value of the importance of an element increases as the physical condition deteriorates. For example, importance of deck concrete with rating 1 is 0.96, whereas its importance is 0.42 when the rating is 9.

#### **4.2 Fuzzification of input data obtained from bridge inspectors**

If Rn is a fuzzy set, representing rating of an element (where 'n' represents rating number i.e. n =0,1,…..9), the general form of the membership function can be formed as follows:

$$\mathbf{R\_n = \mu\_m(r\_m) \mid r\_m} \qquad \quad (\mathbf{m = 0, 1, 2, \dots, 9}) \tag{16}$$

where, (r) is a membership function representing the degree of membership of any fuzzy set and 0 ≤ ≤ 1. The function as described in Eq. (16) quantifies the ambiguity associated with the rating of any element of a bridge. Any rating number can be represented using fuzzy membership function (Emami et al. 1998).


Table 4. Mean values of the structural importance for the deck elements for different rating

Condition Ranking and Rating of Bridges Using Fuzzy Logic 187

**Item 0 1 2 3 4 5 6 7 8 9** 

crack 1 0.94 0.88 0.82 0.76 0.70 0.62 0.51 0.37 0.32

corrosion 1 0.95 0.90 0.84 0.79 0.74 0.66 0.54 0.43 0.36 17 timber decay 1 0.96 0.93 0.89 0.86 0.82 0.72 0.62 0.50 0.44 18 debris, seats 1 0.89 0.78 0.68 0.57 0.46 0.40 0.33 0.25 0.21 19 paint 1 0.90 0.79 0.69 0.58 0.48 0.41 0.34 0.26 0.22

damage 1 0.94 0.87 0.81 0.74 0.68 0.59 0.48 0.34 0.28

Table 6. Mean values of the structural importance for the bridge substructure elements for

R0 = {1.00 0, 0.76 1, 0.55 2, 0.35 3, 0.16 4, 0.00 5, 0.00 6,………. ,0.00 9} and

R9 = {0.00 0, 0.09 1, 0.18 2, 0.28 3, 0.39 4, 0.51 5, 0.62 6, 0.74 7, 0.87 8, 1.00 9}

Fuzzy membership functions for rating values 0 - 9, as obtained above, are shown in Fig. 2.

Rating value =0 Rating value =1 Rating value =2 Rating value =3 Rating value =4 Rating value =5 Rating value =6 Rating value =7 Rating value =8 Rating value =9

0123456789 **Rating value**

R1 = {0.00 0, 1.00 1, 0.45 2, 0.00 3, 0.00 4, 0.00 5, 0.00 6,………. ,0.00 9} Using fuzzy addition, rating membership functions for '2' is calculated as R2 = {0.00 0, 0.45 1, 1.00 2, 0.70 3, 0.45 4, 0.20 5, 0.00 6,………. ,0.00 9}

Usually, the membership values for each rating value are assumed without indication of any specific reason. If membership functions for rating values of 0 and 1 are specified, the membership functions for other rating values can be evaluated using consecutive fuzzy addition rule (Kaufmann & Gupta, 1985). In this study, the rating membership functions for

Mean values of Structural Importance

SL No.

<sup>15</sup>concrete

<sup>20</sup>collision

<sup>16</sup>steel

**Rating**

'0' and '1' are assumed as follows:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 2. Degree of membership of fuzzified rating values

**Degree of membership**


Table 5. Mean values of the structural importance for the superstructure elements for different rating


**Item 0 1 2 3 4 5 6 7 8 9** 

device 1 0.96 0.92 0.07 0.83 0.79 0.71 0.60 0.47 0.42 2 stringers 1 0.96 0.92 0.07 0.83 0.79 0.72 0.61 0.50 0.44 3 girders 1 0.98 0.97 0.95 0.94 0.92 0.85 0.75 0.64 0.58 4 floor beams 1 0.98 0.96 0.94 0.92 0.90 0.83 0.72 0.60 0.54 5 trusses 1 0.97 0.94 0.90 0.87 0.84 0.77 0.67 0.56 0.51 6 paints 1 0.90 0.80 0.70 0.60 0.49 0.43 0.35 0.29 0.24 7 machinery 1 0.94 0.88 0.82 0.76 0.70 0.66 0.58 0.52 0.44 8 rivet or bolts 1 0.96 0.91 0.87 0.82 0.78 0.71 0.61 0.49 0.42 9 weld cracks 1 0.97 0.95 0.92 0.90 0.87 0.83 0.73 0.63 0.56 10 rusts 1 0.95 0.90 0.84 0.79 0.74 0.64 0.54 0.40 0.31 11 timber decay 1 0.97 0.93 0.90 0.86 0.83 0.75 0.65 0.51 0.43

crack 1 0.96 0.91 0.87 0.82 0.78 0.70 0.59 0.49 0.40

damage 1 0.94 0.88 0.83 0.77 0.71 0.64 0.53 0.42 0.35 14 deflection 1 0.95 0.89 0.84 0.78 0.73 0.66 0.59 0.50 0.43 15 alignment 1 0.94 0.88 0.83 0.77 0.71 0.64 0.54 0.44 0.37 16 vibrations 1 0.95 0.88 0.81 0.75 0.69 0.63 0.54 0.43 0.36

**Item 0 1 2 3 4 5 6 7 8 9**  1 bridge seats 1 0.95 0.90 0.86 0.81 0.76 0.68 0.57 0.45 0.40 2 wings 1 0.92 0.73 0.75 0.66 0.58 0.51 0.41 0.33 0.29 3 backwall 1 0.85 0.86 0.80 0.73 0.66 0.58 0.48 0.40 0.35 4 footings 1 0.95 0.90 0.84 0.79 0.74 0.67 0.57 0.46 0.42 5 piles 1 0.94 0.89 0.83 0.78 0.72 0.66 0.56 0.46 0.39 6 erosion 1 0.94 0.87 0.81 0.74 0.68 0.60 0.51 0.40 0.35 7 settlement 1 0.96 0.92 0.87 0.83 0.79 0.70 0.60 0.50 0.45 8 piers, caps 1 0.95 0.89 0.84 0.73 0.78 0.65 0.56 0.46 0.41

columns 1 0.96 0.91 0.87 0.82 0.78 0.70 0.60 0.49 0.43 10 piers, footing 1 0.95 0.90 0.84 0.79 0.74 0.67 0.57 0.47 0.42 11 piers, piles 1 0.95 0.90 0.84 0.79 0.74 0.68 0.59 0.48 0.38 12 piers, scour 1 0.95 0.89 0.84 0.78 0.73 0.65 0.53 0.43 0.45

settlement 1 0.96 0.91 0.87 0.82 0.78 0.70 0.62 0.51 0.42 14 pile, bends 1 0.95 0.90 0.85 0.80 0.75 0.67 0.58 0.48 0.42

Table 5. Mean values of the structural importance for the superstructure elements for

Mean values of Structural Importance

Mean values of Structural Importance

SL No.

<sup>1</sup>bearing

<sup>12</sup>concrete

<sup>13</sup>collision

different rating

<sup>9</sup>piers,

<sup>13</sup>Piers

**Rating**

SL No. **Rating**


Table 6. Mean values of the structural importance for the bridge substructure elements for

Usually, the membership values for each rating value are assumed without indication of any specific reason. If membership functions for rating values of 0 and 1 are specified, the membership functions for other rating values can be evaluated using consecutive fuzzy addition rule (Kaufmann & Gupta, 1985). In this study, the rating membership functions for '0' and '1' are assumed as follows:

R0 = {1.00 0, 0.76 1, 0.55 2, 0.35 3, 0.16 4, 0.00 5, 0.00 6,………. ,0.00 9} and

R1 = {0.00 0, 1.00 1, 0.45 2, 0.00 3, 0.00 4, 0.00 5, 0.00 6,………. ,0.00 9}

Using fuzzy addition, rating membership functions for '2' is calculated as

R2 = {0.00 0, 0.45 1, 1.00 2, 0.70 3, 0.45 4, 0.20 5, 0.00 6,………. ,0.00 9}

R9 = {0.00 0, 0.09 1, 0.18 2, 0.28 3, 0.39 4, 0.51 5, 0.62 6, 0.74 7, 0.87 8, 1.00 9}

Fuzzy membership functions for rating values 0 - 9, as obtained above, are shown in Fig. 2.

Fig. 2. Degree of membership of fuzzified rating values

Condition Ranking and Rating of Bridges Using Fuzzy Logic 189

with the set *A*

The minimum (pessimistic) and maximum (optimistic) values of the intervals for a specific level set correspond respectively to the lower and upper limits of fuzzy membership function at that -level. The set describing the rating of a component at a particular level of

1

*i*

*i*th element at -level. Therefore, the most pessimistic and optimistic range of the resulting set at each -level would form all possible combinations using the discretised non-fuzzy values. Hence, the resolution identity technique provides a convenient way of generalizing

From the above mentioned techniques for processing fuzzy sets, the FWA technique is simpler and faster. As FWA technique does not require discretisation of fuzzy set, accurate result may be achieved with less computational effort provided that the sets representing the rating or importance of different elements are convex. Otherwise, adjustments have to be made to the resulting fuzzy set to ensure its convexity for making the task of transforming a computed fuzzy set into natural language expression easier. Further, another adjustment that is often made to a fuzzy set is the normalization operation to ensure that at least one of the elements of the set contains the degree of membership of one, as suggested by Mullarky and Fenves (1985). On the other hand, accuracy of resolution identity technique depends on refinement of the concerned sets through -level which has a direct impact on computational time. Therefore, in this study, a methodology has been proposed by

In this present approach, the results obtained from eigenvector based priority setting approach combined with FWA for rating of the bridge components are taken as the input for the resolution identity module. It is to be mentioned here that the number of alternatives increase with the increase in the objects (here, components) considered. For example, if a bridge is considered to consist of three main components, such as, 'deck', 'superstructure' and 'substructure' with different ratings and importance factors, number of alternatives produced for each -level is 23+3 = 64 to determine the most optimistic and pessimistic values. Further, for 11 -levels (from 0 to 1.0 in step of 0.1), total number of calculations are

*i n*

 

*R*

is the rating value for the *i*th element at -level, *Wi*

various concepts associated with non-fuzzy sets to fuzzy sets.

judiciously using the advantages of both the techniques.

**4.4 Combined technique for condition rating of existing bridges** 

*n*

1

*i i*

*W R*

 

*W*

*i*

1

(20)

1

0 (or

(21)

is the importance value for the

) is the

0 *A*

, and the symbol

or, A=

1

ranging from 0 and 1.

0 *A*

A=

is the product of a scalar

where,

*A*

> , with

union of the *A*

would be

where, *Ri*

#### **4.3 Overall condition rating of a bridge**

After getting the fuzzified rating and importance of all the elements, it is required to process those sets to arrive at the rating set for the components. In the similar way, the final rating for the bridge can be evaluated by processing the rating and importance sets of components. Generally, the processing of these rating and importance sets is executed using Fuzzy Weighted Average (FWA) or resolution identity technique. Brief details of these two techniques are given below:

#### **4.3.1 Fuzzy Weighted Average (FWA) technique**

Using a structural damage rating scheme according to the local, global and cumulative damage of the structure, resulting damage rating, R, can be evolved (Bertero & Bresler, 1977), using a weighted average approach, as

$$\mathbf{R} = \frac{\sum (w\_i \not\! \mathbf{Q} \mathbf{1}\_i)}{\sum (w\_i \not\! \mathbf{1}\_i \tau\_i)} \tag{17}$$

where, *wi* is the importance factor for the *i*th structural element,


*<sup>i</sup>* is the service history influence coefficient for capacity, and

*<sup>i</sup>* is the resistance (or capacity) in the *i*th element

For a bridge structure, Eq. (2) can be simplified for obtaining the rating as

$$\mathbf{R} = \frac{\sum\_{i=1}^{p} (w\_i \times r\_i)}{\sum\_{i=1}^{p} w\_i} \tag{18}$$

where, *wi* is the importance coefficient of the *i*th object, *ri* is the local rating of the ith object and R is the global or overall rating index when *wi* and *ri* stand for the bridge components, R is the component rating when *wi* and *ri* stand for the bridge elements. Detailed discussions on this methodology are given elsewhere (Sasmal et al., 2004a, 2004b).

#### **4.3.2 Fuzzy resolution identification technique**

A fuzzy set can be easily decomposed into its level sets or intervals through resolution identity as suggested by Dong and Wong (1987). If A is a fuzzy set of universe (U), then an -level set or alpha cut of A is a non-fuzzy set denoted by A which comprises of all elements of U whose grade of membership in A is greater than or equal to .

A can be expressed in symbolic form as:

$$\mathbf{A}\_a = \{ \mathbf{u} \mid \mu\_\mathbf{A}(\mathbf{u}) \ge \alpha \} \tag{19}$$

In mathematical form, the fuzzy set A can be decomposed into its level sets through the resolution identity such that

188 Fuzzy Logic – Emerging Technologies and Applications

After getting the fuzzified rating and importance of all the elements, it is required to process those sets to arrive at the rating set for the components. In the similar way, the final rating for the bridge can be evaluated by processing the rating and importance sets of components. Generally, the processing of these rating and importance sets is executed using Fuzzy Weighted Average (FWA) or resolution identity technique. Brief details of these two

Using a structural damage rating scheme according to the local, global and cumulative damage of the structure, resulting damage rating, R, can be evolved (Bertero & Bresler,

> *w w*

( ) *ii i iii*

 

(17)

(18)

**4.3 Overall condition rating of a bridge** 

**4.3.1 Fuzzy Weighted Average (FWA) technique** 

R= ( )

*<sup>i</sup>* is the resistance (or capacity) in the *i*th element

**4.3.2 Fuzzy resolution identification technique** 

A can be expressed in symbolic form as:

resolution identity such that

where, *wi* is the importance factor for the *i*th structural element,

*<sup>i</sup>* is the service history influence coefficient for capacity, and

*<sup>i</sup>* is the service history coefficient for structural response (or demand), *i* is the structural response (or demand) in the *i*th element due to load,

For a bridge structure, Eq. (2) can be simplified for obtaining the rating as

on this methodology are given elsewhere (Sasmal et al., 2004a, 2004b).

elements of U whose grade of membership in A is greater than or equal to .

R = 1

*i p i i*

*p*

1

where, *wi* is the importance coefficient of the *i*th object, *ri* is the local rating of the ith object and R is the global or overall rating index when *wi* and *ri* stand for the bridge components, R is the component rating when *wi* and *ri* stand for the bridge elements. Detailed discussions

A fuzzy set can be easily decomposed into its level sets or intervals through resolution identity as suggested by Dong and Wong (1987). If A is a fuzzy set of universe (U), then an -level set or alpha cut of A is a non-fuzzy set denoted by A which comprises of all

 A = {u A(u) } (19) In mathematical form, the fuzzy set A can be decomposed into its level sets through the

( )

*w r*

*w*

*i i*

1977), using a weighted average approach, as

techniques are given below:

$$\mathbf{A} = \sum\_{\alpha=0}^{1} \alpha A\_{\alpha} \quad \text{or, } \mathbf{A} = \underset{\mathbf{0}}{\int} \alpha A\_{\alpha} \tag{20}$$

where, *A* is the product of a scalar with the set *A* , and the symbol 1 0 (or ) is the union of the *A* , with ranging from 0 and 1.

The minimum (pessimistic) and maximum (optimistic) values of the intervals for a specific level set correspond respectively to the lower and upper limits of fuzzy membership

function at that -level. The set describing the rating of a component at a particular level of would be

$$\mathcal{R}\_{\alpha} = \frac{\sum\_{i=1}^{n} \mathcal{W}\_{i\alpha} \mathcal{R}\_{i\alpha}}{\sum\_{i=1}^{n} \mathcal{W}\_{i\alpha}} \tag{21}$$

where, *Ri*is the rating value for the *i*th element at -level, *Wi* is the importance value for the *i*th element at -level. Therefore, the most pessimistic and optimistic range of the resulting set at each -level would form all possible combinations using the discretised non-fuzzy values. Hence, the resolution identity technique provides a convenient way of generalizing various concepts associated with non-fuzzy sets to fuzzy sets.

From the above mentioned techniques for processing fuzzy sets, the FWA technique is simpler and faster. As FWA technique does not require discretisation of fuzzy set, accurate result may be achieved with less computational effort provided that the sets representing the rating or importance of different elements are convex. Otherwise, adjustments have to be made to the resulting fuzzy set to ensure its convexity for making the task of transforming a computed fuzzy set into natural language expression easier. Further, another adjustment that is often made to a fuzzy set is the normalization operation to ensure that at least one of the elements of the set contains the degree of membership of one, as suggested by Mullarky and Fenves (1985). On the other hand, accuracy of resolution identity technique depends on refinement of the concerned sets through -level which has a direct impact on computational time. Therefore, in this study, a methodology has been proposed by judiciously using the advantages of both the techniques.

#### **4.4 Combined technique for condition rating of existing bridges**

In this present approach, the results obtained from eigenvector based priority setting approach combined with FWA for rating of the bridge components are taken as the input for the resolution identity module. It is to be mentioned here that the number of alternatives increase with the increase in the objects (here, components) considered. For example, if a bridge is considered to consist of three main components, such as, 'deck', 'superstructure' and 'substructure' with different ratings and importance factors, number of alternatives produced for each -level is 23+3 = 64 to determine the most optimistic and pessimistic values. Further, for 11 -levels (from 0 to 1.0 in step of 0.1), total number of calculations are

Condition Ranking and Rating of Bridges Using Fuzzy Logic 191

 Li = [i,1(c1)c1, i,2(c2)c2, i,3(c3)c3, , i,n(cn)cn ]i = 1, p (22) in which (ci) is the membership grade corresponding to alternative ci. The solution would be the optimal alternative which has the highest degree of acceptability with respect to all relevant goals Li. Towards this, several models have been introduced, in recent years, for fuzzy MADM but the eigenvector based priority setting approach is considered as one of the

For the general MADM problem described using Eq. (22), a positive, non-zero number in the priority vector (W) corresponding to each object indicates the power of importance (i) of that object in the decision process. By applying the associated powers 1, 2, 3,…. p to the

The decision function D is then obtained from the intersection of the fuzzy sets representing

where, D1(c1), D2(c2), D3(c3), ……………, Dn(cn) are the decision values corresponding to the

The final decision is the one that corresponds to maximum of all decision values. Hence, the

 Dfinal = max[Dj(cj) cj]; j = 1,n (27) For the bridge rating application, let e1, e2, e3,……,ep be the elements considered under each component of the bridge. The fuzzy set for a given condition rating ri of element ei can be

Li = [R]ei = {i,1 r1, i,2 r2,…….., i,9 r9} (28)

Therefore, decision value (D) for rating of any element can be evaluated from Eq. 25 as

The final rating of each of the major bridge components is found from the decision values as

Dfinal = max { d1 r1, d2 r2, ……………,d9 r9}

2,j <sup>j</sup> [μ (c ) c2, <sup>α</sup><sup>3</sup>

i,3 <sup>3</sup> [<sup>μ</sup> (c ) c3, ……… <sup>α</sup><sup>i</sup>

………….. L *<sup>p</sup>*

3,j <sup>j</sup> [<sup>μ</sup> (c ) c3, ………, <sup>α</sup><sup>p</sup>

D = [d1 r1, d2 r2, d3 r3, ……………, dn rn] (29)

Or, D = [D1(c1) c1, D2(c2) c2, D3(c3) c3, ……………, Dn(cn) cn] (25)

*p* 

i,n <sup>n</sup> [μ (c ) cn]; i = 1, p (23)

(24)

p,j <sup>j</sup> [μ (c ) cp]; j = 1, n (26)

**4.6 Application of priority vector in MADM model for condition rating** 

α α i i L [ i i,1 <sup>μ</sup> <sup>1</sup> (c ) c1, <sup>α</sup><sup>i</sup>

<sup>α</sup><sup>1</sup> D ( ) min[μ1,j <sup>j</sup> (c ) *j j <sup>c</sup>* c1, <sup>α</sup><sup>2</sup>

expressed as the objective (goal) Li, as

fuzzy objective sets L1, L2, L 3,….., L p respectively, the following can be obtained:

i,2 <sup>2</sup> [<sup>μ</sup> (c ) c2, <sup>α</sup><sup>i</sup>

D = 1 L1 <sup>2</sup> L2 <sup>3</sup> L3 

alternatives c1, c2,c3,…..,cn, and are given by Aturaliya (1994) as :

where i = 1 , p and i,n is the membership value of element ei at rn.

best alternatives.

the goals as

final decision becomes

704. The same increases enormously with the increase in number of objects (here, components or elements of the bridge). If the procedure mentioned above is implemented for a bridge which is assumed to be divided into 10 components, the number of calculations required to get the resultant rating fuzzy set would be 210+10 11 = 11534336. In fact, the availability of high speed microcomputers has made this approach attractive and practical for actual bridge inspection, management and planning applications.

Further, question may arise that why the resolution identity technique alone can not be applied for the whole bridge rating system by avoiding FWA technique. The simple answer is that component rating can be evolved by the simple FWA technique because there is no need for tackling non-convexity of the assigned sets unless it is essential. Otherwise, for the whole rating evaluation, number of calculations would be enormous. For a bridge having 3 components with 13, 16 and 20 elements (as described in Table 3) under the components, the total number of calculations required for the final result using resolution identity alone would be in the order of 1.21013. Hence, a combined technique is proposed in this study to get the accurate result without much increase in computing time.

In the approach proposed in this study, priority setting values of elements are calculated to evaluate the power of importance of each element in describing the condition of a particular component. The usual techniques available for condition rating combine the rating and importance of elements to arrive at the rating of each component. But, the importance factor, as mentioned earlier, is very much dependent on the prevailing condition (rating) of the particular element. Thus, a minor element with worse condition may unnecessarily reduce the rating value of that component under which the element is grouped. This problem can be tackled by the introduction of power of importance which is independent of the prevailing condition of elements. As mentioned earlier, imprecision, subjective judgment and uncertainty are associated with bridge inspection data. Because of uncertainty, the bridge inspector may not exactly know the prevailing condition (rating) of a particular element of a bridge. Moreover, importance factor for an element depends on its rating. But, decision on rating is a difficult proposition. Under these circumstances, several models were introduced for decision making in a fuzzy environment. In this study, Multi-Attributive Decision Making (MADM) model has been adopted as a decision tool.

#### **4.5 Multi-Attributive Decision Making (MADM) model**

Multi-Attributive Decision Making (MADM) model is one of the methods in decision studies where the factors towards a priority decision are many (multi-criteria). The assessment of bridge rating can be viewed as a Multi-Attributive Decision Making model because of its many components and sub-components (elements). In this study, an attempt has been made to use MADM model, to get the priority vector of elements depending on their importance over the others which would lead to a reliable decision (rating) from the bridge inspection data. The general MADM model can be expressed as follows:

Let L = {Lii = 1,2,3,….,p} be a set of goals and C = {cjj=1,2,3,….,n} be a finite set of decision alternatives from which the acceptability of the alternatives is judged. The objective is to select the one, from these alternatives, that best satisfies the set of goals, L1,…….Lp. The objective function can be expressed in the form of fuzzy set as follows:

704. The same increases enormously with the increase in number of objects (here, components or elements of the bridge). If the procedure mentioned above is implemented for a bridge which is assumed to be divided into 10 components, the number of calculations required to get the resultant rating fuzzy set would be 210+10 11 = 11534336. In fact, the availability of high speed microcomputers has made this approach attractive and practical

Further, question may arise that why the resolution identity technique alone can not be applied for the whole bridge rating system by avoiding FWA technique. The simple answer is that component rating can be evolved by the simple FWA technique because there is no need for tackling non-convexity of the assigned sets unless it is essential. Otherwise, for the whole rating evaluation, number of calculations would be enormous. For a bridge having 3 components with 13, 16 and 20 elements (as described in Table 3) under the components, the total number of calculations required for the final result using resolution identity alone would be in the order of 1.21013. Hence, a combined technique is proposed in this study to

In the approach proposed in this study, priority setting values of elements are calculated to evaluate the power of importance of each element in describing the condition of a particular component. The usual techniques available for condition rating combine the rating and importance of elements to arrive at the rating of each component. But, the importance factor, as mentioned earlier, is very much dependent on the prevailing condition (rating) of the particular element. Thus, a minor element with worse condition may unnecessarily reduce the rating value of that component under which the element is grouped. This problem can be tackled by the introduction of power of importance which is independent of the prevailing condition of elements. As mentioned earlier, imprecision, subjective judgment and uncertainty are associated with bridge inspection data. Because of uncertainty, the bridge inspector may not exactly know the prevailing condition (rating) of a particular element of a bridge. Moreover, importance factor for an element depends on its rating. But, decision on rating is a difficult proposition. Under these circumstances, several models were introduced for decision making in a fuzzy environment. In this study, Multi-Attributive

Multi-Attributive Decision Making (MADM) model is one of the methods in decision studies where the factors towards a priority decision are many (multi-criteria). The assessment of bridge rating can be viewed as a Multi-Attributive Decision Making model because of its many components and sub-components (elements). In this study, an attempt has been made to use MADM model, to get the priority vector of elements depending on their importance over the others which would lead to a reliable decision (rating) from the

Let L = {Lii = 1,2,3,….,p} be a set of goals and C = {cjj=1,2,3,….,n} be a finite set of decision alternatives from which the acceptability of the alternatives is judged. The objective is to select the one, from these alternatives, that best satisfies the set of goals, L1,…….Lp. The

bridge inspection data. The general MADM model can be expressed as follows:

objective function can be expressed in the form of fuzzy set as follows:

for actual bridge inspection, management and planning applications.

get the accurate result without much increase in computing time.

Decision Making (MADM) model has been adopted as a decision tool.

**4.5 Multi-Attributive Decision Making (MADM) model** 

$$\mathbf{L}\_{i} = \begin{bmatrix} \mu\_{\nu 1}(\mathbf{c}\_{1}) | \mathbf{c}\_{1\prime} \ \mu\_{\nu 2}(\mathbf{c}\_{2}) | \mathbf{c}\_{2\prime} \ \mu\_{\nu 3}(\mathbf{c}\_{3}) | \mathbf{c}\_{3\prime} \ \mu\_{\nu \nu}(\mathbf{c}\_{n}) | \mathbf{c}\_{n} \end{bmatrix} \tag{22}$$

in which (ci) is the membership grade corresponding to alternative ci. The solution would be the optimal alternative which has the highest degree of acceptability with respect to all relevant goals Li. Towards this, several models have been introduced, in recent years, for fuzzy MADM but the eigenvector based priority setting approach is considered as one of the best alternatives.

#### **4.6 Application of priority vector in MADM model for condition rating**

For the general MADM problem described using Eq. (22), a positive, non-zero number in the priority vector (W) corresponding to each object indicates the power of importance (i) of that object in the decision process. By applying the associated powers 1, 2, 3,…. p to the fuzzy objective sets L1, L2, L 3,….., L p respectively, the following can be obtained:

$$\mathbf{L}\_{\mathbf{i}}^{\mathbf{a}\_{\mathbf{i}}} = \left[ \boldsymbol{\mu}\_{\mathbf{i},1}^{\mathbf{a}\_{\mathbf{i}}}(\mathbf{c}\_{1}) \mid \mathbf{c}\_{1} \quad \left[ \boldsymbol{\mu}\_{\mathbf{i},2}^{\mathbf{a}\_{\mathbf{i}}}(\mathbf{c}\_{2}) \mid \mathbf{c}\_{2} \quad \left[ \boldsymbol{\mu}\_{\mathbf{i},3}^{\mathbf{a}\_{\mathbf{i}}}(\mathbf{c}\_{3}) \mid \mathbf{c}\_{3} \dots \dots \dots \quad \left[ \boldsymbol{\mu}\_{\mathbf{i},n}^{\mathbf{a}\_{\mathbf{i}}}(\mathbf{c}\_{n}) \mid \mathbf{c}\_{n} \right] \right] \right] \tag{23}$$

The decision function D is then obtained from the intersection of the fuzzy sets representing the goals as

$$\mathbf{D} = \mathbf{L}\_1^{a\_1} \cap \mathbf{L}\_2^{a\_2} \cap \mathbf{L}\_3^{a\_3} \text{ ...} \dots \text{....} \dots \text{//} \mathbf{L}\_p^{a\_p} \tag{24}$$

$$\text{Or}, \text{D} = \left[ \text{D}\_{\text{l}}(\text{c}\_{\text{l}}) \mid \text{c}\_{\text{l}} \text{ D}\_{\text{2}}(\text{c}\_{\text{2}}) \mid \text{c}\_{\text{2}} \text{ D} \text{(c}\_{\text{3}}) \mid \text{c}\_{\text{3}} \text{ } \dots \text{ } \dots \dots \dots \text{ } \text{D}\_{\text{n}}(\text{c}\_{\text{n}}) \mid \text{c}\_{\text{n}} \right] \tag{25}$$

where, D1(c1), D2(c2), D3(c3), ……………, Dn(cn) are the decision values corresponding to the alternatives c1, c2,c3,…..,cn, and are given by Aturaliya (1994) as :

$$\mathbf{D}\_{\rangle}(\mathbf{c}\_{\rangle}) = \min \{ \mu\_{1\not\downarrow}^{a\_{\sharp}}(\mathbf{c}\_{\not\slash}) \mid \mathbf{c}\_{\not\succ} \quad \| \mu\_{2\not\succ}^{a\_{\sharp}}(\mathbf{c}\_{\not\succ}) \mid \mathbf{c}\_{\not\succ} \quad \| \mu\_{3\not\succ}^{a\_{\sharp}}(\mathbf{c}\_{\not\succ}) \mid \mathbf{c}\_{\not\succ} , \dots, \dots, \ \| \mu\_{p\not\succ}^{a\_{p}}(\mathbf{c}\_{\not\succ}) \mid \mathbf{c}\_{\not\succ} \} \colon\_{\mathbf{1},\mathbf{n}} \tag{26}$$

The final decision is the one that corresponds to maximum of all decision values. Hence, the final decision becomes

$$\mathbf{D}\_{\text{final}} = \max \{ \mathbf{D}\_{\bar{l}}(\mathbf{c}\_{\bar{l}}) \mid \mathbf{c}\_{\bar{l}} \}; \quad \mathbf{j} = \mathbf{1}, \mathbf{n} \tag{27}$$

For the bridge rating application, let e1, e2, e3,……,ep be the elements considered under each component of the bridge. The fuzzy set for a given condition rating ri of element ei can be expressed as the objective (goal) Li, as

$$\mathbf{L}\_{i} = \begin{bmatrix} \mathbf{R} \end{bmatrix}\_{ei} = \begin{Bmatrix} \mu\_{i,1} \mid \mathbf{r}\_{1\prime} \ \mu\_{i,2} \mid \mathbf{r}\_{2\prime}, \dots, \dots, \mu\_{i,9} \mid \mathbf{r}\_{9} \end{Bmatrix} \tag{28}$$

where i = 1 , p and i,n is the membership value of element ei at rn.

Therefore, decision value (D) for rating of any element can be evaluated from Eq. 25 as

$$\mathbf{D} = \begin{bmatrix} \mathbf{d}\_1 \mid \mathbf{r}\_1, \mathbf{d}\_2 \mid \mathbf{r}\_2 \; \mathbf{d}\_3 \mid \mathbf{r}\_3 & \dots & \dots & \dots & \mathbf{d}\_n \mid \mathbf{r}\_n \end{bmatrix} \tag{29}$$

The final rating of each of the major bridge components is found from the decision values as

$$\mathbf{D}\_{\text{final}} = \max\left\{ |\mathbf{d}\_1| \, \mathbf{r}\_1, \mathbf{d}\_2| \, \mathbf{r}\_2, \dots, \mathbf{r}, \dots, \mathbf{d}\_\bullet \, | \, \mathbf{r}\_\bullet \right\}$$

Condition Ranking and Rating of Bridges Using Fuzzy Logic 193

evaluated through visual inspection are grouped under visual assessment. The items which need a detailed inspection, comprehensive observation and thorough study are grouped under general assessment. Further, the items which would require rigorous testing, both at site and in the laboratory using sophisticated instrumentation are grouped under detailed assessment. The comparison matrices for different components are formed (Triantaphyllou et al., 1997; Sasmal et al. 2006). The relative importance of any item with respect to the other, under any layer, may change depending on location, societal importance and decision objectives. The eigen solutions of the comparison matrices are carried out for different criteria layers to check for the consistency (in other words, check for acceptance of the comparison matrix) of the elements assigned for different items under criteria layer. The largest eigenvalue (max) of the comparison matrix corresponding to each layer is obtained by solving Eq. 6 and this eigenvalue is used for calculating the consistency ratio (CR). The values for consistency index (CI) are obtained by using Eq. 7 and the consistency ratio (CR) is obtained by dividing CI with random consistency index (RCI). The values of max, CI, RCI and CR for different items under criteria layer are presented in Table 7 and the values of CR for different items (index layers) under criteria layer are within the acceptable limit (<10%). Hence, the comparison matrix assigned for index layers are accepted for further study.

Visual assessment (16) 17.3951 0.0930 1.56 5.962 General assessment (15) 16.1770 0.0841 1.56 5.391 Detailed assessment (14) 15.1707 0.0901 1.56 5.776

**5.2 Calculation of relative weights of different index layers on criteria layer** 

assignment of the weights for formation of comparison matrix using AHP.

procedures. Hence, the comparison matrix has to be modified accordingly.

**5.3 Formulation of higher layer comparison matrix using AHP** 

The relative weights for components of index layers (i), (j) and (k) are established using the Eq. 3. The relative importance (weights) of items under index layers (i, j and k) obtained in this study are presented in Table 8. From the table, it is clear that there are considerable differences in relative weights of items in each index layer which signify their importance on the functionality of a bridge as a whole. It is also worthy to mention here that a large variation of relative weights of items signifies the necessity for correct, logical and realistic

Next step is to form the pair-wise comparison matrix for criteria layer to get the optimum goal in objective layer. Following the scheme described above, relative weights of each item of the criteria layer has to be evaluated. Table 8 shows both the comparison matrix between criteria layers and the relative weights of each criteria layer on objective layer. It signifies that the relative weight of detailed assessment on condition assessment of existing bridge is much more than that of the visual assessment. But, it may be noted that the relative weights of different assessments on condition assessment of overall bridge may change with the type of bridge, specific site condition and the degree of accuracy of different assessment

Table 7. Check for consistency of pair-wise assigned weights

max CI =(max-n)/(n-1) RCI CR = (CI/RCI)100%

= [dm rm] (30)

Hence, the rating of the particular component would be 'm' that represents any integer value between 1 and 9.

#### **5. A case study for illustration of the proposed methodology**

Computer programs have been developed based on the formulations presented in the preceding sections for condition evaluation of existing bridges and rating of the most deserved one. Based on the formulation discussed in the previous sections and the computer program developed in this study using the formulations, a study has been made for priority ranking of bridges. The data corresponding to five RC bridges (Br1, Br2, Br3, Br4 and Br5) has been adopted. In order to use the AHP to rank these bridges, at first an Analytic Hierarchy Model (AHM) with three layers, such as, objective layer (OL), criteria layer (CL) and index layer (IL) is constructed, as shown in Fig. 3. This hierarchy model is constructed by the authors based on the information available from FHWA (1991) and Liang et al. (2003) to demonstrate the proposed methodology. It is worthy to mention that the proposed methodology can be used for any hierarchy model. Therefore, it may be noted that the appropriate item(s) under any layer (as shown in Fig. 3) may be added or deleted depending on the requirement for assessing the condition of concerned bridges.

Fig. 3. Analytic Hierarchy Model (AHM) for Existing RC Bridge

After establishing the model, a set of relative importance (weights) between each single factor evaluation (item) is set-up for controlling the reliability of layer ranking. Combination of relative importance (weights) to each single item forms a comparison matrix. Condition evaluation of the considered bridges through prioritization and the rating of the most deserved bridge are arrived using the methodology described in the preceding sections. The whole procedure has been described in following sections for better illustration and understanding.

#### **5.1 Formulation of comparison matrices for each layer and check for consistency**

The first step is to carry out pair-wise comparisons of items under each layer of the AHP model as shown in Fig. 3. In this study, criteria layer is divided into three parts, namely, visual assessment, general assessment and detailed assessment. The items which can be

Hence, the rating of the particular component would be 'm' that represents any integer

Computer programs have been developed based on the formulations presented in the preceding sections for condition evaluation of existing bridges and rating of the most deserved one. Based on the formulation discussed in the previous sections and the computer program developed in this study using the formulations, a study has been made for priority ranking of bridges. The data corresponding to five RC bridges (Br1, Br2, Br3, Br4 and Br5) has been adopted. In order to use the AHP to rank these bridges, at first an Analytic Hierarchy Model (AHM) with three layers, such as, objective layer (OL), criteria layer (CL) and index layer (IL) is constructed, as shown in Fig. 3. This hierarchy model is constructed by the authors based on the information available from FHWA (1991) and Liang et al. (2003) to demonstrate the proposed methodology. It is worthy to mention that the proposed methodology can be used for any hierarchy model. Therefore, it may be noted that the appropriate item(s) under any layer (as shown in Fig. 3) may be added or deleted

**5. A case study for illustration of the proposed methodology** 

depending on the requirement for assessing the condition of concerned bridges.

Assessment of Existing Bridge

Visual Assessment (i) General Assessment (j) Detailed Assessment (k)

After establishing the model, a set of relative importance (weights) between each single factor evaluation (item) is set-up for controlling the reliability of layer ranking. Combination of relative importance (weights) to each single item forms a comparison matrix. Condition evaluation of the considered bridges through prioritization and the rating of the most deserved bridge are arrived using the methodology described in the preceding sections. The whole procedure has been described in following sections for better illustration and

**5.1 Formulation of comparison matrices for each layer and check for consistency** 

The first step is to carry out pair-wise comparisons of items under each layer of the AHP model as shown in Fig. 3. In this study, criteria layer is divided into three parts, namely, visual assessment, general assessment and detailed assessment. The items which can be

1. slab 2. girders 3. …. 4…… …….. 15. abutment

Fig. 3. Analytic Hierarchy Model (AHM) for Existing RC Bridge

value between 1 and 9.

1. wearing surface 2. deck condition

16. Bearing device

3. …… 4 ……. ……….

understanding.

= [dm rm] (30)

1. existing prestress 2. concrete strength

Objective Layer (OB)

Criteria Layer (CR)

Index Layer (ID)

3. ….. 4. ….. ……… 14.anchor block evaluated through visual inspection are grouped under visual assessment. The items which need a detailed inspection, comprehensive observation and thorough study are grouped under general assessment. Further, the items which would require rigorous testing, both at site and in the laboratory using sophisticated instrumentation are grouped under detailed assessment. The comparison matrices for different components are formed (Triantaphyllou et al., 1997; Sasmal et al. 2006). The relative importance of any item with respect to the other, under any layer, may change depending on location, societal importance and decision objectives. The eigen solutions of the comparison matrices are carried out for different criteria layers to check for the consistency (in other words, check for acceptance of the comparison matrix) of the elements assigned for different items under criteria layer. The largest eigenvalue (max) of the comparison matrix corresponding to each layer is obtained by solving Eq. 6 and this eigenvalue is used for calculating the consistency ratio (CR). The values for consistency index (CI) are obtained by using Eq. 7 and the consistency ratio (CR) is obtained by dividing CI with random consistency index (RCI). The values of max, CI, RCI and CR for different items under criteria layer are presented in Table 7 and the values of CR for different items (index layers) under criteria layer are within the acceptable limit (<10%). Hence, the comparison matrix assigned for index layers are accepted for further study.


Table 7. Check for consistency of pair-wise assigned weights

#### **5.2 Calculation of relative weights of different index layers on criteria layer**

The relative weights for components of index layers (i), (j) and (k) are established using the Eq. 3. The relative importance (weights) of items under index layers (i, j and k) obtained in this study are presented in Table 8. From the table, it is clear that there are considerable differences in relative weights of items in each index layer which signify their importance on the functionality of a bridge as a whole. It is also worthy to mention here that a large variation of relative weights of items signifies the necessity for correct, logical and realistic assignment of the weights for formation of comparison matrix using AHP.

#### **5.3 Formulation of higher layer comparison matrix using AHP**

Next step is to form the pair-wise comparison matrix for criteria layer to get the optimum goal in objective layer. Following the scheme described above, relative weights of each item of the criteria layer has to be evaluated. Table 8 shows both the comparison matrix between criteria layers and the relative weights of each criteria layer on objective layer. It signifies that the relative weight of detailed assessment on condition assessment of existing bridge is much more than that of the visual assessment. But, it may be noted that the relative weights of different assessments on condition assessment of overall bridge may change with the type of bridge, specific site condition and the degree of accuracy of different assessment procedures. Hence, the comparison matrix has to be modified accordingly.

Condition Ranking and Rating of Bridges Using Fuzzy Logic 195

**Item weight** 

**bridge** 

3 kerbs 0.0152 0 2 2 1 0 4 median 0.0591 0 4 0 0 2 5 sidewalks 0.0189 0 2 2 2 0 6 parapets 0.0134 2 0 1 1 0 7 railing 0.0439 4 2 0 0 0 8 paints 0.0179 3 4 2 2 1 9 drains 0.1442 6 6 4 6 0 10 lighting 0.0282 9 9 0 0 0 11 utilities 0.0645 2 2 4 4 6 12 joint leakage 0.2012 0 0 3 9 12

16 others 0.0431 12 9 2 4 9

2 girders 0.0165 27 18 0 18 6 3 slab beams 0.0421 4 trusses 0.0252

6 rivet bolts 0.0704

9 erosion 0.0483 12 18 2 18 3

11 pier 0.1563 18 0 12 3 9 12 pier shaft 0.0375 0 12 0 0 0 13 friction layer 0.0867 0 9 0 0 0 14 abutment 0.2081 12 6 9 3 2 15 others 0.1900 0 2 3 12 9

0.297 1 stringers 0.0338

 **Estimation indices of items of** 

Br1 Br2 Br3 Br4 Br5

0.0342 0 0 0 2 1

0.0194 2 0 12 9 3

0.1881 1 4 4 6 2

0.0719 3 3 12 18 3

0.0367 4 2 0 0 2

0.0200 12 0 0 2 9

0.0250 36 12 36 2 9

0.0169 27 27 12 18 3

0.0233 18 36 0 3 27

**Estimation items** 

surface

condition

13 expansion joint

masonry

14 bearing device

5 chloride content

7 concrete crack

settlement

10 substructure protection

8 pier

15 wing

2 deck

**Estimation Criterion** 

Visual assessment (i)

General assessment (j) **Subsystem weight** 

**Item No.** 

0.126 1 wearing


Table 8. Relative weights of items under index layer on criteria layer

#### **5.4 Fuzzy synthesis and evaluation of membership functions**

This step deals with the assessment of condition of bridge items under index layer (in this case, i, j and k). In this study, the assessment of items has been carried out by determining the estimation indices of the items as described in preceeding section. The estimation indices of the items for different bridges (Br1, Br2, Br3, Br4 and Br5) are presented in Table 9. In this table, the relative weights of components, are calculated using the procedure described above. In this study, the optimistic membership evaluation function has been used for developing membership functions. Using the estimation indices of items as tabulated in Table 9, the membership degrees, Rn,m (n=1 to 3; m = 1 to 16/15/14) of each items are calculated for the bridges (Br1, Br2, Br3, Br4, Br5) considered for assessment.

#### **5.5 Fuzzy synthesis evaluation matrix and priority ranking values**

The relationship between the membership degrees Rn,m (in which n= 1,2….no of criteria layers, and m=1,2,…no of items in each index layer) of each single factor (alternative) evaluation index and weight, *Wn* , is *D WR n n nm* . as per Eq. (14), where the value *Dn* in Eq. (14) is the fuzzy synthesis evaluation matrix. The proposition of *Dn* is to construct the membership function for each alternative of evaluation set. Based on the fuzzy mathematics theory, the fuzzy synthesis evaluation result, *B* , of any factor can be expressed as in Eq. 15.

In this case, *B* = 0.456636 0.256391 0.296120 0.525126 0.43876

**Index layer (j) General assessment** 

**Item Relative weight Item Relative weight Item Relative weight**  i1 0.0342 j1 0.0338 k1 0.0233 i2 0.0194 j2 0.0165 k2 0.0572 i3 0.0152 j3 0.0421 k3 0.0225 i4 0.0591 j4 0.0252 k4 0.1373 i5 0.0189 j5 0.0200 k5 0.0298 i6 0.0134 j6 0.0704 k6 0.0684 i7 0.0439 j7 0.0250 k7 0.1875 i8 0.0179 j8 0.0169 k8 0.1816 i9 0.1442 j9 0.0483 k9 0.0840 i10 0.0282 j10 0.0233 k10 0.0457 i11 0.0645 j11 0.1563 k11 0.0425 i12 0.2012 j12 0.0375 k12 0.0369 i13 0.1881 j13 0.0867 k13 0.0212 i14 0.0719 i14 0.2081 k14 0.0621

This step deals with the assessment of condition of bridge items under index layer (in this case, i, j and k). In this study, the assessment of items has been carried out by determining the estimation indices of the items as described in preceeding section. The estimation indices of the items for different bridges (Br1, Br2, Br3, Br4 and Br5) are presented in Table 9. In this table, the relative weights of components, are calculated using the procedure described above. In this study, the optimistic membership evaluation function has been used for developing membership functions. Using the estimation indices of items as tabulated in Table 9, the membership degrees, Rn,m (n=1 to 3; m = 1 to 16/15/14) of each items are

The relationship between the membership degrees Rn,m (in which n= 1,2….no of criteria layers, and m=1,2,…no of items in each index layer) of each single factor (alternative) evaluation index and weight, *Wn* , is *D WR n n nm* . as per Eq. (14), where the value *Dn* in Eq. (14) is the fuzzy synthesis evaluation matrix. The proposition of *Dn* is to construct the membership function for each alternative of evaluation set. Based on the fuzzy mathematics theory, the fuzzy synthesis evaluation result, *B* , of any factor can be expressed as in Eq. 15.

**Index layer (k) Detailed assessment** 

**Index layer (i) Visual assessment** 

i16 0.0431

i15 0.0367 j15 0.1900

Table 8. Relative weights of items under index layer on criteria layer

calculated for the bridges (Br1, Br2, Br3, Br4, Br5) considered for assessment.

In this case, *B* = 0.456636 0.256391 0.296120 0.525126 0.43876

**5.5 Fuzzy synthesis evaluation matrix and priority ranking values** 

**5.4 Fuzzy synthesis and evaluation of membership functions** 


Condition Ranking and Rating of Bridges Using Fuzzy Logic 197

reported in (Aturaliya, 1994). Fuzzy sets for rating of the components, i.e, deck, superstructure and substructure are shown in Table 10 and corresponding importances

> **Br1 Br2 Br3 Br4 Br5 Bridges to be prioritized**

 **0 1 2 3 4 5 6 7 8 9**  1. Bridge Deck 0.00 0.33 0.67 1.00 0.83 0.67 0.50 0.33 0.17 0.00 2. Superstructure 0.00 0.25 0.50 0.75 1.00 0.88 0.75 0.63 0.50 0.38 3. Substructure 0.00 0.25 0.50 0.75 1.00 0.88 0.75 0.63 0.50 0.38

 **0 1 2 3 4 5 6 7 8 9**  1. Bridge Deck 1.00 1.00 0.90 0.80 0.70 0.61 0.51 0.45 0.33 0.23 2. Superstructure 1.00 1.00 0.96 0.92 0.87 0.83 0.79 0.71 0.60 0.47 3. Substructure 1.00 1.00 0.80 0.70 0.60 0.50 0.35 0.25 0.15 0.10

As described earlier, the resolution identity technique is adopted in this study to get the final rating of the bridge when the component ratings (from the elemental values) are computed using eigenvector based priority setting technique using MADM combined with FWA method. Hence, for arriving at the final rating of the most deserved bridge, the basic data considered are the calculated ratings of the components and computed weights as shown in Tables 8 and 9. The fuzzy membership functions of rating and weights of different components (deck, superstructure and substructure) thus obtained, are discretised using resolution identity technique. Here, each set is discretised into 11 -levels (from 0.0 to 1.0 in

Fig. 4. Priority ranking values of the bridges considered for condition assessment

Table 10. Computed fuzzified rating values for different components of the bridge

Components Rating membership

Components Importance factors

Table 11. Importance membership functions of the components

(weights) are shown in Table 11.

0.00

0.10

0.20

0.30

**Priority Vector**

0.40

0.50

0.60


Represents the non-availability of estimation data

Table 9. Estimation indices of items of the bridges considered for condition assessment

The fuzzy synthesis evaluation result, *B* , actually shows the relative condition of existing bridges considered. Therefore, the values under *B* can also be treated as the priority vector for condition assessment of the bridges. As the optimistic membership evaluation function is used in this study (given in Eq. 10), the higher value in fuzzy synthesis evaluation result for a bridge in comparison to the other ones signifies greater degree of distress. In this study, the condition of Br4 among the five bridges considered here can be treated as most deficient and similarly, Br2 would be the best. The condition priority order of the bridges considered here for illustration is shown in Fig. 4. From the figure, it may be noted that the priority order of the bridges considered is as follows:

#### =[Br4, Br1, Br5, Br3, Br2]

#### **5.6 Condition rating of the most deserved bridge**

Using the fuzzy mathematics, ratings of different component of the bridge, Br4, are calculated using FWA. Importance(weight) of different elements has been considered as

**Item weight** 

**bridge** 

4 vibration 0.1373 18 9 0 36 27

7 deflection 0.1875 12 18 36 48 12 8 footing 0.1816 18 0 3 12 6

10 piles 0.0457 11 pier-column 0.0425 12 2 9 6 9 12 pier footing 0.0369 18 3 12 3 18

14 anchor block 0.0621 27 12 0 36 48

 **Estimation indices of items of** 

Br1 Br2 Br3 Br4 Br5

0.0233 36 12 0 36 27

0.0572 27 18 36 36 48

0.0225 12 0 12 18 9

0.0298 18 0 0 0 18

0.0684 0 36 0 0 12

0.0840 0 0 0 0 12

0.0212 36 12 0 18 36

**Estimation items** 

prestress

2 concrete strength

3 foundation mat

5 prevention earthquake block

corrosion

6 steel

9 collision damage

13 prestressing cable corrosion

Table 9. Estimation indices of items of the bridges considered for condition assessment

The fuzzy synthesis evaluation result, *B* , actually shows the relative condition of existing bridges considered. Therefore, the values under *B* can also be treated as the priority vector for condition assessment of the bridges. As the optimistic membership evaluation function is used in this study (given in Eq. 10), the higher value in fuzzy synthesis evaluation result for a bridge in comparison to the other ones signifies greater degree of distress. In this study, the condition of Br4 among the five bridges considered here can be treated as most deficient and similarly, Br2 would be the best. The condition priority order of the bridges considered here for illustration is shown in Fig. 4. From the figure, it may be noted that the priority

=[Br4, Br1, Br5, Br3, Br2]

Using the fuzzy mathematics, ratings of different component of the bridge, Br4, are calculated using FWA. Importance(weight) of different elements has been considered as

Represents the non-availability of estimation data

order of the bridges considered is as follows:

**5.6 Condition rating of the most deserved bridge** 

**Estimation Criterion** 

Detailed assessment (k) **Subsystem weight** 

**Item No.** 

0.577 1 existing

reported in (Aturaliya, 1994). Fuzzy sets for rating of the components, i.e, deck, superstructure and substructure are shown in Table 10 and corresponding importances (weights) are shown in Table 11.

Fig. 4. Priority ranking values of the bridges considered for condition assessment




Table 11. Importance membership functions of the components

As described earlier, the resolution identity technique is adopted in this study to get the final rating of the bridge when the component ratings (from the elemental values) are computed using eigenvector based priority setting technique using MADM combined with FWA method. Hence, for arriving at the final rating of the most deserved bridge, the basic data considered are the calculated ratings of the components and computed weights as shown in Tables 8 and 9. The fuzzy membership functions of rating and weights of different components (deck, superstructure and substructure) thus obtained, are discretised using resolution identity technique. Here, each set is discretised into 11 -levels (from 0.0 to 1.0 in

Condition Ranking and Rating of Bridges Using Fuzzy Logic 199

 In view of this, a methodology based on AHP has been used, in this chapter, for ranking of the existing bridges towards their assessment of prevailing condition which would

 The comparison matrices for different layers of hierarchy are formulated for arriving at the relative weights of the items under each layer. An eigen solution is carried out for each comparison matrix to extract the largest eigenvalue which is further used for checking the consistency of the formulation of the comparison matrices. Estimation indices of individual bridge components have to be arrived based on the bridge inspector's observation and the results of field and laboratory testing. Thus, the estimation indices would suffer from subjective judgement and uncertainty. Hence, an optimistic fuzzy membership function has been used to scale the indices of all the components of the bridges uniformly. Based on the fuzzy synthesized evaluation

 For evaluating the condition rating of the most deserved bridge determined from the prioritization, it is found that as the number of elements of bridge components increase the complexity in arriving at a unique rating number using Fuzzy Weighted Average (FWA) also increases. Hence, a resolution identity method is incorporated in the methodology to take care of the problems that may arise due to non-convexity and

 Further, for the component rating, the Multi-Attributive Decision Making (MADM) model based on priority vector of the constituent elements of the component is also considered because it gives a more realistic representation of the condition of the

 A computer programs have been developed based on the proposed methodology for condition evaluation through prioritization and rating of bridges. It is found that the methodology is capable of handling any number of bridges without any limitation on consideration of components, and elements and rating scale. Thus, the proposed methodology would certainly help the engineers and policy makers concerned with bridge management to arrive at a systematic judgment and to formulate methodical

steps towards retrofitting, rehabilitation or demolition of bridge in future years. It is worthy to mention here that though the condition evaluation through fuzzy logic based AHP may be used as an useful tool for decision making, it should be utilised with adequate care because the whole procedure is dependent on different estimation indices of controlling parameters which have to be taken from inspector's observation

Aturaliya, S.P. (1994). *Bridge condition rating based on fuzzy set theory and Eigenvector approach*,

Ben-Arieh, D. & Triantaphyllou, E. (1992). Quantifying data for group technology with weighted fuzzy features, *Int. J. of Production Research,* Vol. 30, pp. 1285-1299. Bertero, V.V. & Bresler, B. (1977). Design and engineering decisions: Failure criteria (Limit

Masters thesis, Department of Civil Engineering, Kansas State University,

State): Developing methodologies for Evaluating the earth quake safety of Existing

matrix, the priority ranking of the bridges has been evolved.

requirement of normalisation of the concerned sets.

and results of field and laboratory testing.

Manhattan, Kansas.

help in fixing their repair order.

component.

**7. References** 

step of 0.1). For better illustration, the resolution identification of the fuzzy set representing the rating of the deck component of the bridge concerned is shown in Fig. 5. Further discussion can be found elsewhere [Sasmal el al. (2005)].

Fig. 5. Resolution identification of the fuzzy set representing rating of the deck component

At each -level, there would be 64 combinations to get the most optimistic (maximum) and pessimistic (minimum) range of the fuzzy set at that -level. For 11 -levels (as considered in this study) the optimistic and pessimistic ranges of the resultant set and the membership representation of the resultant rating (RR) derived from the pessimistic and optimistic ranges using resolution identity technique is shown in Fig. 5. The resultant rating of the bridge, as a whole, has been defuzzified using MATLAB, to get the rating value of the bridge. For this particular case, the defuzzification has been executed using the centroidal method and the rating value is obtained as '**4.6668**'. From the result, it is clear that the rating of the bridge (Br4) falls in between 4 and 5 but closer to 5. It may be the decision maker's discretion in considering the rating value depending on the practical condition and other factors like the environmental condition, importance of the bridge as a whole on the societal service etc. As mentioned earlier, a scale of 0-9 has been considered in this study. So, the condition of the bridge (Br4) falls between 4 and 5 which perhaps reflect the moderate condition. Since, the condition rating of the most deserved bridge among the bridges considered in this study is in between 4 and 5, hence the other bridges are comparatively in better condition.

#### **6. Concluding remarks**

 In bridge engineering, systematic identification of the order of degree of deficiency of the bridges that are considered for their condition assessment is a usual problem. Until now, no systematic approach seems to be available for priority ranking of existing bridges.

step of 0.1). For better illustration, the resolution identification of the fuzzy set representing the rating of the deck component of the bridge concerned is shown in Fig. 5. Further

Fig. 5. Resolution identification of the fuzzy set representing rating of the deck component

At each -level, there would be 64 combinations to get the most optimistic (maximum) and pessimistic (minimum) range of the fuzzy set at that -level. For 11 -levels (as considered in this study) the optimistic and pessimistic ranges of the resultant set and the membership representation of the resultant rating (RR) derived from the pessimistic and optimistic ranges using resolution identity technique is shown in Fig. 5. The resultant rating of the bridge, as a whole, has been defuzzified using MATLAB, to get the rating value of the bridge. For this particular case, the defuzzification has been executed using the centroidal method and the rating value is obtained as '**4.6668**'. From the result, it is clear that the rating of the bridge (Br4) falls in between 4 and 5 but closer to 5. It may be the decision maker's discretion in considering the rating value depending on the practical condition and other factors like the environmental condition, importance of the bridge as a whole on the societal service etc. As mentioned earlier, a scale of 0-9 has been considered in this study. So, the condition of the bridge (Br4) falls between 4 and 5 which perhaps reflect the moderate condition. Since, the condition rating of the most deserved bridge among the bridges considered in this study is in between 4 and 5, hence the other bridges are comparatively in

 In bridge engineering, systematic identification of the order of degree of deficiency of the bridges that are considered for their condition assessment is a usual problem. Until now, no systematic approach seems to be available for priority ranking of existing

discussion can be found elsewhere [Sasmal el al. (2005)].

better condition.

bridges.

**6. Concluding remarks** 


#### **7. References**


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**0**

**10**

*Japan*

**Fuzzy Logic for Multi-Hop Broadcast in**

A Vehicular Ad hoc Network (VANET) is a form of mobile ad hoc network in which vehicles are equipped with wireless communication devices. Vehicular ad hoc networks have been attracting the interest of both academic and industrial communities on account of their important role in Intelligent Transportation Systems (ITS). VANETs are expected to be able to significantly reduce the number of road accidents. When vehicles travel at a high speed on roads, drivers have very little time to react to the vehicle in front of them. By using vehicular ad hoc networks, emergency information can be propagated along the road to notify drivers ahead of time so that necessary actions can be taken to avoid accidents. Vehicular ad hoc networks also make the driving more efficient by disseminating traffic warning information

In this chapter, we consider VANET broadcast protocols which work as a basis of many vehicular applications especially safety applications. Providing reliable and efficient multi-hop broadcast in vehicular ad hoc networks is very challenging. First, in vehicular ad hoc networks, vehicles are usually deployed in a dense manner. Therefore, a simple broadcast scheme cannot work well because of redundant broadcasts. Second, wireless communications are unreliable and vehicles can move at a high speed. Consequently, it is difficult to reduce

As a solution, we explain an approach which uses a fuzzy logic to enhance multi-hop broadcast in vehicular ad hoc networks. Due to the high node density, vehicle movement and fading feature of wireless communications, providing a reliable and efficient multi-hop broadcast in vehicular ad hoc networks is still an open research topic. Using only a subset of neighbor nodes to relay broadcast messages is a main concept for providing efficiency. Meanwhile, in order to ensure a high reliability, multiple metrics of inter-vehicle distance, node mobility and signal strength should be jointly considered in the relay node selection. However, these metrics conflict with each other and these conflicts depend on the vehicle mobility, vehicle distribution and fading condition. The mathematical model of the optimal relay problem is complex to derive and a solution based on it would be too expensive for practical application. Therefore, we employ fuzzy logic to handle these imprecise and uncertain information. We use a fuzzy logic based method to select relay nodes by jointly considering inter-vehicle distance, node mobility and signal strength. The selected relay nodes can provide a reliable data forwarding with a high efficiency. In this chapter, we give a detailed

the redundant broadcast while maintaining a high packet dissemination ratio.

description of the fuzzy logic based method with simulation results.

**1. Introduction**

and service information.

Celimuge Wu, Satoshi Ohzahata and Toshihiko Kato

**Vehicular Ad Hoc Networks**

*University of Electro-Communications*

Zadeh, L.A. (1973). The concept of a linguistic variable and its application in approximate reasoning, *Information Science, part I*: Vol. 8, No. 3, pp. 199-249.

Zadeh L.A. (1965). Fuzzy sets, *Information & Control*, Vol. 8, pp. 338-353.


### **Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks**

Celimuge Wu, Satoshi Ohzahata and Toshihiko Kato *University of Electro-Communications Japan*

#### **1. Introduction**

202 Fuzzy Logic – Emerging Technologies and Applications

Zadeh, L.A. (1973). The concept of a linguistic variable and its application in approximate

Zhao, Z. & Chen C. (2002). A Fuzzy system for concrete bridge damage diagnosis, *Computers* 

Zahedi, F. (1986). The analytic hierarchy process – a survey of the method and its

reasoning, *Information Science, part I*: Vol. 8, No. 3, pp. 199-249.

Zadeh L.A. (1965). Fuzzy sets, *Information & Control*, Vol. 8, pp. 338-353.

applications, *Interfaces*, Vol. 16, No. 4, pp. 96-108.

*and Structures*, Vol. 80, pp. 629-641.

A Vehicular Ad hoc Network (VANET) is a form of mobile ad hoc network in which vehicles are equipped with wireless communication devices. Vehicular ad hoc networks have been attracting the interest of both academic and industrial communities on account of their important role in Intelligent Transportation Systems (ITS). VANETs are expected to be able to significantly reduce the number of road accidents. When vehicles travel at a high speed on roads, drivers have very little time to react to the vehicle in front of them. By using vehicular ad hoc networks, emergency information can be propagated along the road to notify drivers ahead of time so that necessary actions can be taken to avoid accidents. Vehicular ad hoc networks also make the driving more efficient by disseminating traffic warning information and service information.

In this chapter, we consider VANET broadcast protocols which work as a basis of many vehicular applications especially safety applications. Providing reliable and efficient multi-hop broadcast in vehicular ad hoc networks is very challenging. First, in vehicular ad hoc networks, vehicles are usually deployed in a dense manner. Therefore, a simple broadcast scheme cannot work well because of redundant broadcasts. Second, wireless communications are unreliable and vehicles can move at a high speed. Consequently, it is difficult to reduce the redundant broadcast while maintaining a high packet dissemination ratio.

As a solution, we explain an approach which uses a fuzzy logic to enhance multi-hop broadcast in vehicular ad hoc networks. Due to the high node density, vehicle movement and fading feature of wireless communications, providing a reliable and efficient multi-hop broadcast in vehicular ad hoc networks is still an open research topic. Using only a subset of neighbor nodes to relay broadcast messages is a main concept for providing efficiency. Meanwhile, in order to ensure a high reliability, multiple metrics of inter-vehicle distance, node mobility and signal strength should be jointly considered in the relay node selection. However, these metrics conflict with each other and these conflicts depend on the vehicle mobility, vehicle distribution and fading condition. The mathematical model of the optimal relay problem is complex to derive and a solution based on it would be too expensive for practical application. Therefore, we employ fuzzy logic to handle these imprecise and uncertain information. We use a fuzzy logic based method to select relay nodes by jointly considering inter-vehicle distance, node mobility and signal strength. The selected relay nodes can provide a reliable data forwarding with a high efficiency. In this chapter, we give a detailed description of the fuzzy logic based method with simulation results.

as

where *Ns* is the number of slots.

**2.2 Sender-oriented protocols**

node movement.

in the relay node selection.

**2.2.1 MPR**

probability *p*. Otherwise, the node discards the packet.

packet loss would occur at the neighbor node.

*TStr* = *Str* × *τ*, (2)

*<sup>R</sup>* )�, (3)

where *τ* is the estimated one-hop delay, which includes the medium access delay and

Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks 205

*Str* <sup>=</sup> �*Ns*(<sup>1</sup> <sup>−</sup> *min*(*Dtr*, *<sup>R</sup>*)

Similar to slotted 1-persistence scheme, in the slotted *p*-persistence scheme, upon reception of a packet, a node checks the packet ID. If the node receives the packet only once in the assigned time slot *TStr* which is calculated as Eq. (2), the node rebroadcasts with the predetermined

In the sender-oriented protocols, since the sender node specifies relay nodes, the redundant broadcasts can be minimized. The relay node selection method directly affects the performance of a sender-oriented protocol. Generally, the relay node selection is based on the information collected from the exchange of hello messages. Qayyum et al. (2002) have proposed a multipoint relay (MPR) broadcast scheme (here we call MPR Broadcast) in which relay nodes are selected using two-hop neighbor information. Djedid et al. (2008) have proposed a broadcast protocol which selects relay nodes based on Connected Dominating Set. However, these protocols do not consider node mobility in the relay node selection. As a result, the selected relay node can become sub-optimal and can lose the message due to the

In our previous work (Wu et al. (2010)), we have proposed a relay node selection which considers the additional radio coverage and node movement (here we call EMPR Broadcast). However, EMPR Broadcast does not consider the fading feature of wireless channels. In a wireless channel, a node can receive a hello message from a neighbor which is at a distance where stable communication is impossible. If the neighbor node is selected as a relay node, a

Sahoo et al. (2009) have proposed BPAB, a Binary Partition Assisted emergency Broadcast protocol for vehicular Ad hoc networks. BPAB intends to use the farthest node to relay messages. However, in a fading channel, the farthest node can lose the messages. Therefore, we have to choose the nodes which have stable signal strength as relay nodes. In short, multiple metrics of inter-vehicle distance, mobility and signal strength should be considered

Qayyum et al. (2002) have proposed a multipoint relay (MPR) broadcast scheme (here we call MPR Broadcast). MPR can substantially reduce the message overhead as compared to the flooding. In MPR broadcast, each node selects a set of its neighbor nodes as "multipoint relays" (MPR). Only the selected MPR nodes are responsible for forwarding the messages. The neighbors of node *N* which are not in its MPR set, receive and process broadcast messages but do not retransmit broadcast messages received from node *N*. MPR broadcast provides an efficient mechanism for disseminating messages by reducing the number of transmissions.

propagation delay. *Str* is the assigned slot number, which is calculated by

The basic idea of the approach has been published by IEEE (Wu et al. (2010)). However, in this chapter, we use a more realistic model to evaluate the approach and present our new simulation results. We explain the approach with new and more detailed information.

#### **2. Multi-hop broadcast in vehicular ad hoc networks**

The simplest way to disseminate information is flooding. In the flooding, each node rebroadcasts a packet upon the first reception. Obviously, in a high-density network, the flooding introduces too many redundant broadcasts and consequently incurs collisions and results in a low dissemination rate. There have been a lot of protocols to reduce the redundant broadcasts in a high-density network. These protocols can be classified into two categories of sender-oriented protocols and receiver-oriented protocols. In the sender-oriented protocols, a sender node specifies relay nodes. In contrast, in the receiver-oriented protocols, upon reception of a message, a receiver node determines own action (whether rebroadcast the message or not) in an autonomous manner.

#### **2.1 Receiver-oriented protocols**

Several receiver based broadcast protocols have been proposed. Wisitpongphan & Tonguz (2007) have proposed three broadcast schemes: weighted *p*-persistence, slotted 1-persistence, and slotted *p*-persistence schemes. In these protocols, upon reception of a message, a node calculates a broadcast probability according to the distance from the sender node. Generally, a larger distance from the sender node results in a higher broadcast probability. Suriyapaiboonwattana et al. (2009) have proposed a protocol which uses an adaptive wait time and adaptive probability to trigger the rebroadcast. Slavik & Mahgoub (2010) have proposed a protocol in which all nodes rebroadcast a received message with a certain probability. Mylonas et al. (2008) have proposed a Speed Adaptive Probabilistic Flooding algorithm to determine the rebroadcast probability according to vehicle speed. However, in the receiver-based protocols, each node determines whether rebroadcast or not in an autonomous manner. Therefore, redundant broadcasts cannot be eliminated entirely.

#### **2.1.1 Weighted** *p***-persistence, slotted 1-persistence and slotted** *p***-persistence scheme**

Wisitpongphan & Tonguz (2007) have proposed three probabilistic and timer-based broadcast suppression techniques. They are weighted *p*-persistence, slotted 1-persistence and slotted *p*-persistence Scheme.

In the weighted *p*-persistence scheme, upon reception of a packet from node *t*, node *r* checks the packet ID and rebroadcasts with probability *ptr* if node *r* receives the packet for the first time. Otherwise, the node discards the packet. The probability, *ptr*, is calculated on a per packet basis using

$$p\_{tr} = \frac{D\_{tr}}{R} \,\prime \tag{1}$$

where *Dtr* is the relative distance between nodes *t* and *r*, *R* is the average transmission range. The larger the *Dtr*, the higher the probability will be.

In slotted 1-persistence scheme, upon reception of a packet, a node checks the packet ID. If the node receives the packet for the first time and fails to detect any rebroadcast from other nodes in an assigned time slot *TStr*, the node rebroadcasts the packet. If the node can detect a rebroadcast of the packet from any other nodes, the node discards the packet. *TStr* is calculated as

2 Will-be-set-by-IN-TECH

The basic idea of the approach has been published by IEEE (Wu et al. (2010)). However, in this chapter, we use a more realistic model to evaluate the approach and present our new simulation results. We explain the approach with new and more detailed information.

The simplest way to disseminate information is flooding. In the flooding, each node rebroadcasts a packet upon the first reception. Obviously, in a high-density network, the flooding introduces too many redundant broadcasts and consequently incurs collisions and results in a low dissemination rate. There have been a lot of protocols to reduce the redundant broadcasts in a high-density network. These protocols can be classified into two categories of sender-oriented protocols and receiver-oriented protocols. In the sender-oriented protocols, a sender node specifies relay nodes. In contrast, in the receiver-oriented protocols, upon reception of a message, a receiver node determines own action (whether rebroadcast the

Several receiver based broadcast protocols have been proposed. Wisitpongphan & Tonguz (2007) have proposed three broadcast schemes: weighted *p*-persistence, slotted 1-persistence, and slotted *p*-persistence schemes. In these protocols, upon reception of a message, a node calculates a broadcast probability according to the distance from the sender node. Generally, a larger distance from the sender node results in a higher broadcast probability. Suriyapaiboonwattana et al. (2009) have proposed a protocol which uses an adaptive wait time and adaptive probability to trigger the rebroadcast. Slavik & Mahgoub (2010) have proposed a protocol in which all nodes rebroadcast a received message with a certain probability. Mylonas et al. (2008) have proposed a Speed Adaptive Probabilistic Flooding algorithm to determine the rebroadcast probability according to vehicle speed. However, in the receiver-based protocols, each node determines whether rebroadcast or not in an

autonomous manner. Therefore, redundant broadcasts cannot be eliminated entirely.

**2.1.1 Weighted** *p***-persistence, slotted 1-persistence and slotted** *p***-persistence scheme**

Wisitpongphan & Tonguz (2007) have proposed three probabilistic and timer-based broadcast suppression techniques. They are weighted *p*-persistence, slotted 1-persistence and slotted

In the weighted *p*-persistence scheme, upon reception of a packet from node *t*, node *r* checks the packet ID and rebroadcasts with probability *ptr* if node *r* receives the packet for the first time. Otherwise, the node discards the packet. The probability, *ptr*, is calculated on a per

*ptr* <sup>=</sup> *Dtr*

where *Dtr* is the relative distance between nodes *t* and *r*, *R* is the average transmission range.

In slotted 1-persistence scheme, upon reception of a packet, a node checks the packet ID. If the node receives the packet for the first time and fails to detect any rebroadcast from other nodes in an assigned time slot *TStr*, the node rebroadcasts the packet. If the node can detect a rebroadcast of the packet from any other nodes, the node discards the packet. *TStr* is calculated

*<sup>R</sup>* , (1)

**2. Multi-hop broadcast in vehicular ad hoc networks**

message or not) in an autonomous manner.

**2.1 Receiver-oriented protocols**

*p*-persistence Scheme.

packet basis using

The larger the *Dtr*, the higher the probability will be.

$$T\_{\mathbb{S}\_{tr}} = \mathbb{S}\_{tr} \times \tau\_{\prime} \tag{2}$$

where *τ* is the estimated one-hop delay, which includes the medium access delay and propagation delay. *Str* is the assigned slot number, which is calculated by

$$\mathcal{S}\_{tr} = \lceil \mathcal{N}\_{\mathbf{s}} (1 - \frac{\min\{D\_{tr}, \mathcal{R}\}}{\mathcal{R}}) \rceil\,. \tag{3}$$

where *Ns* is the number of slots.

Similar to slotted 1-persistence scheme, in the slotted *p*-persistence scheme, upon reception of a packet, a node checks the packet ID. If the node receives the packet only once in the assigned time slot *TStr* which is calculated as Eq. (2), the node rebroadcasts with the predetermined probability *p*. Otherwise, the node discards the packet.

#### **2.2 Sender-oriented protocols**

In the sender-oriented protocols, since the sender node specifies relay nodes, the redundant broadcasts can be minimized. The relay node selection method directly affects the performance of a sender-oriented protocol. Generally, the relay node selection is based on the information collected from the exchange of hello messages. Qayyum et al. (2002) have proposed a multipoint relay (MPR) broadcast scheme (here we call MPR Broadcast) in which relay nodes are selected using two-hop neighbor information. Djedid et al. (2008) have proposed a broadcast protocol which selects relay nodes based on Connected Dominating Set. However, these protocols do not consider node mobility in the relay node selection. As a result, the selected relay node can become sub-optimal and can lose the message due to the node movement.

In our previous work (Wu et al. (2010)), we have proposed a relay node selection which considers the additional radio coverage and node movement (here we call EMPR Broadcast). However, EMPR Broadcast does not consider the fading feature of wireless channels. In a wireless channel, a node can receive a hello message from a neighbor which is at a distance where stable communication is impossible. If the neighbor node is selected as a relay node, a packet loss would occur at the neighbor node.

Sahoo et al. (2009) have proposed BPAB, a Binary Partition Assisted emergency Broadcast protocol for vehicular Ad hoc networks. BPAB intends to use the farthest node to relay messages. However, in a fading channel, the farthest node can lose the messages. Therefore, we have to choose the nodes which have stable signal strength as relay nodes. In short, multiple metrics of inter-vehicle distance, mobility and signal strength should be considered in the relay node selection.

#### **2.2.1 MPR**

Qayyum et al. (2002) have proposed a multipoint relay (MPR) broadcast scheme (here we call MPR Broadcast). MPR can substantially reduce the message overhead as compared to the flooding. In MPR broadcast, each node selects a set of its neighbor nodes as "multipoint relays" (MPR). Only the selected MPR nodes are responsible for forwarding the messages. The neighbors of node *N* which are not in its MPR set, receive and process broadcast messages but do not retransmit broadcast messages received from node *N*. MPR broadcast provides an efficient mechanism for disseminating messages by reducing the number of transmissions.

where *i* − 1 indicates the previous value (the value is updated on the reception of a hello message). Eq. (6) could give a larger value for the same directed vehicles and smaller value for vehicles that moving toward opposite direction. If a node *x* has opposite moving direction to the sender, corresponding *θ* will be smaller than other vehicles which have the same direction

Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks 207

Upon reception of a hello from its neighbor, a sender node updates a neighbor's *PMF* as

Every node maintains a *PMF* (*PMFi*−1(*x*)) and *AC* (*ACi*−1(*x*)) for every one-hop neighbor. In Eq. (7), the *PMFi*−1(*x*) is initialized to 0. Similarly, *ACi*−1(*x*) is initialized to *<sup>φ</sup>* in Eq. (6). The sender node uses these values, the current *MF* (*MFi*(*x*)) and *AC* (*ACi*(*x*)) to calculate the latest *PMF* (*PMFi*(*x*)) as shown in Eq. (6) and Eq. (7). The node then updates the *PMFi*−1(*x*) and *ACi*−1(*x*). *PMF*(*x*) is reset to zero if the sender fails to hear any hello message from node

In Ref. (Wu et al. (2010)), a retransmission method also has been proposed. However, in this

Receiver-oriented approaches cannot reduce the redundant broadcasts entirely. As a result, it is difficult to guarantee a high data dissemination ratio. In this chapter we consider using a sender-oriented approach. However, in the sender-oriented approach, when a relay node fails to receive a packet, the data delivery fails. Therefore, selecting efficient and reliable relay

In vehicular ad hoc networks, redundant rebroadcasts incur packet collisions and a higher end-to-end delay due to the increase of MAC layer contention time. It is important to reduce the broadcast redundancy by selecting a small subset of nodes to relay a broadcast packet. However, the relay node selection uses the information collected from the exchange of hello messages. In a highly mobile network, the selected relay node can move out the transmission range of the sender node. Moreover, a node can receive a hello message from a neighbor which is at a distance where stable communication is impossible. If an inappropriate neighbor node

Therefore, in the relay node selection, multiple metrics of inter-vehicle distance, node mobility and signal strength should be considered jointly. However, it is difficult to establish a satisfactory relay node evaluation criterion for the following reasons. First, the network information (inter-vehicle distance, node mobility and signal strength) known by each node is inaccurate, incomplete and imprecise. Second, since these metrics may conflict with each

As shown in Fig. 1, if we select the farthest node as a relay node, it minimizes the number of relays (efficiency up). But that relay node may lose the packet because the signal is weak (reliability down). Moreover, due to the node movement, the relay node might move out the transmission range of the sender node. These conflicts depend on the vehicle mobility, vehicle distribution and fading condition. Therefore, the mathematical model of the optimal relay problem is complex to derive and a solution based on it would be

*PMFi*(*x*) ← (<sup>1</sup> − *<sup>μ</sup>*)*PMFi*−1(*x*) + *<sup>μ</sup>* × *<sup>θ</sup>* × *MFi*(*x*). (7)

because its additional radio coverage (*AC*(.)) is changing frequently.

follows.

**2.3 Challenges**

**3. Why fuzzy logic**

other, it results in uncertainty.

*x* in three times the hello interval.

chapter, we do not consider the retransmission issue.

nodes is the most important issue for sender-oriented protocols.

is selected as a relay node, the neighbor node fails to receive the message.

Every node attaches its one hop neighbors to the hello messages. In this way, every node is aware of its two-hop neighbors. Each node selects its MPR set from its one-hop neighbors. This set is selected such that these nodes cover (in terms of radio range) all two-hop neighbor nodes. The MPR set of *N*, denoted as *MPR*(*N*), is then an arbitrary subset of the one-hop neighbor of *N*. *MPR*(*N*) satisfies the following condition: every node in the two-hop neighborhood of *N* must have a link towards *MPR*(*N*). The smaller a MPR set (in term of the number of nodes in the set), the less the message overhead.

The following is a heuristic for the selection of MPR nodes.


MPR can optimize the message dissemination by minimizing the number of messages flooded in the network. The technique is particularly suitable for large and dense networks. However, MPR cannot be used in vehicular ad hoc networks without enhancement because MPR does not consider node mobility at all. In vehicular ad hoc networks, because of node movement, the neighbor information can be imprecise, resulting in the selected relay nodes fail to receive the packets.

#### **2.2.2 EMPR**

In addition to the radio coverage, EMPR (Wu et al. (2010)) considers node mobility in the relay node selection. EMPR algorithm introduces predicted MPR fitness (*PMF*) to evaluate a node whether it is suitable for relaying broadcast packet or not. A sender node selects the neighbor which has the maximal *PMF* as a relay node from the possible candidate nodes.

Upon reception of a hello message from node *x*, sender node *s* calculates the corresponding multipoint relay fitness (*MF*(*x*)) as

$$MF\_i(\mathbf{x}) = \frac{|AC\_i(\mathbf{x})|}{|N\_i(\mathbf{s}) \cup N\_i(\mathbf{x})|} \tag{4}$$

where *i* indicates the current value. *Ni*(*x*) denotes neighbor set of node *x*, |*Ni*(*x*)| denotes number of *x*'s one hop neighbors. *AC*(*x*) is defined as

$$A\mathcal{C}(\mathbf{x}) = \overline{N(\mathbf{s})} \cap N(\mathbf{x}).\tag{5}$$

Eq. (4) could give a higher value for a node that has larger additional radio coverage.

In order to provide different weights to different level of movements, EMPR algorithm introduces discount rate *θ* which is calculated as

$$\theta = \begin{cases} \sqrt{\frac{|A\mathcal{C}\_i(\mathbf{x}) \cap A\mathcal{C}\_{i-1}(\mathbf{x})|}{|A\mathcal{C}\_i(\mathbf{x}) \cup A\mathcal{C}\_{i-1}(\mathbf{x})|}} & \text{if } A\mathcal{C}\_i(\mathbf{x}) \cup A\mathcal{C}\_{i-1}(\mathbf{x}) \neq \phi \\ 0, & \text{otherwise}, \end{cases} \tag{6}$$

where *i* − 1 indicates the previous value (the value is updated on the reception of a hello message). Eq. (6) could give a larger value for the same directed vehicles and smaller value for vehicles that moving toward opposite direction. If a node *x* has opposite moving direction to the sender, corresponding *θ* will be smaller than other vehicles which have the same direction because its additional radio coverage (*AC*(.)) is changing frequently.

Upon reception of a hello from its neighbor, a sender node updates a neighbor's *PMF* as follows.

$$PMF\_i(\mathbf{x}) \leftarrow (1 - \mu)PMF\_{i-1}(\mathbf{x}) + \mu \times \theta \times MF\_i(\mathbf{x}).\tag{7}$$

Every node maintains a *PMF* (*PMFi*−1(*x*)) and *AC* (*ACi*−1(*x*)) for every one-hop neighbor. In Eq. (7), the *PMFi*−1(*x*) is initialized to 0. Similarly, *ACi*−1(*x*) is initialized to *<sup>φ</sup>* in Eq. (6). The sender node uses these values, the current *MF* (*MFi*(*x*)) and *AC* (*ACi*(*x*)) to calculate the latest *PMF* (*PMFi*(*x*)) as shown in Eq. (6) and Eq. (7). The node then updates the *PMFi*−1(*x*) and *ACi*−1(*x*). *PMF*(*x*) is reset to zero if the sender fails to hear any hello message from node *x* in three times the hello interval.

In Ref. (Wu et al. (2010)), a retransmission method also has been proposed. However, in this chapter, we do not consider the retransmission issue.

#### **2.3 Challenges**

4 Will-be-set-by-IN-TECH

Every node attaches its one hop neighbors to the hello messages. In this way, every node is aware of its two-hop neighbors. Each node selects its MPR set from its one-hop neighbors. This set is selected such that these nodes cover (in terms of radio range) all two-hop neighbor nodes. The MPR set of *N*, denoted as *MPR*(*N*), is then an arbitrary subset of the one-hop neighbor of *N*. *MPR*(*N*) satisfies the following condition: every node in the two-hop neighborhood of *N* must have a link towards *MPR*(*N*). The smaller a MPR set (in term of

2. First select those one-hop neighbor nodes in *N*(*x*) as multipoint relays which are the only neighbor of some node in *N*2(*x*), and add these one-hop neighbor nodes to the multipoint

**(a)** For each node in *N*(*x*) which is not in *MPR*(*x*), compute the number of nodes that the

MPR can optimize the message dissemination by minimizing the number of messages flooded in the network. The technique is particularly suitable for large and dense networks. However, MPR cannot be used in vehicular ad hoc networks without enhancement because MPR does not consider node mobility at all. In vehicular ad hoc networks, because of node movement, the neighbor information can be imprecise, resulting in the selected relay nodes fail to receive

In addition to the radio coverage, EMPR (Wu et al. (2010)) considers node mobility in the relay node selection. EMPR algorithm introduces predicted MPR fitness (*PMF*) to evaluate a node whether it is suitable for relaying broadcast packet or not. A sender node selects the neighbor

Upon reception of a hello message from node *x*, sender node *s* calculates the corresponding

<sup>|</sup>*Ni*(*s*) <sup>∪</sup> *Ni*(*x*)<sup>|</sup> (4)

(6)

*AC*(*x*) = *N*(*s*) ∩ *N*(*x*). (5)

, if *ACi*(*x*) ∪ *ACi*−1(*x*) �= *<sup>φ</sup>*

*MFi*(*x*) = <sup>|</sup>*ACi*(*x*)<sup>|</sup>

Eq. (4) could give a higher value for a node that has larger additional radio coverage.

0, otherwise,

�|*ACi*(*x*)∩*ACi*−<sup>1</sup>(*x*)<sup>|</sup> |*ACi*(*x*)∪*ACi*−<sup>1</sup>(*x*)|

where *i* indicates the current value. *Ni*(*x*) denotes neighbor set of node *x*, |*Ni*(*x*)| denotes

In order to provide different weights to different level of movements, EMPR algorithm

which has the maximal *PMF* as a relay node from the possible candidate nodes.

3. While there still exist some node in *N*2(*x*) which is not covered by *MPR*(*x*):

the number of nodes in the set), the less the message overhead. The following is a heuristic for the selection of MPR nodes.

node covers among the uncovered nodes in the set *N*2(*x*).

**(b)** Add the node which has the maximal this number to *MPR*(*x*).

1. Start with an empty multipoint relay set *MPR*(*x*).

relay set *MPR*(*x*).

the packets.

**2.2.2 EMPR**

multipoint relay fitness (*MF*(*x*)) as

number of *x*'s one hop neighbors. *AC*(*x*) is defined as

introduces discount rate *θ* which is calculated as

⎧ ⎨ ⎩

*θ* =

Receiver-oriented approaches cannot reduce the redundant broadcasts entirely. As a result, it is difficult to guarantee a high data dissemination ratio. In this chapter we consider using a sender-oriented approach. However, in the sender-oriented approach, when a relay node fails to receive a packet, the data delivery fails. Therefore, selecting efficient and reliable relay nodes is the most important issue for sender-oriented protocols.

#### **3. Why fuzzy logic**

In vehicular ad hoc networks, redundant rebroadcasts incur packet collisions and a higher end-to-end delay due to the increase of MAC layer contention time. It is important to reduce the broadcast redundancy by selecting a small subset of nodes to relay a broadcast packet. However, the relay node selection uses the information collected from the exchange of hello messages. In a highly mobile network, the selected relay node can move out the transmission range of the sender node. Moreover, a node can receive a hello message from a neighbor which is at a distance where stable communication is impossible. If an inappropriate neighbor node is selected as a relay node, the neighbor node fails to receive the message.

Therefore, in the relay node selection, multiple metrics of inter-vehicle distance, node mobility and signal strength should be considered jointly. However, it is difficult to establish a satisfactory relay node evaluation criterion for the following reasons. First, the network information (inter-vehicle distance, node mobility and signal strength) known by each node is inaccurate, incomplete and imprecise. Second, since these metrics may conflict with each other, it results in uncertainty.

As shown in Fig. 1, if we select the farthest node as a relay node, it minimizes the number of relays (efficiency up). But that relay node may lose the packet because the signal is weak (reliability down). Moreover, due to the node movement, the relay node might move out the transmission range of the sender node. These conflicts depend on the vehicle mobility, vehicle distribution and fading condition. Therefore, the mathematical model of the optimal relay problem is complex to derive and a solution based on it would be

the hello interval. The hello interval is set to 1 second. Before broadcasting a packet, a sender node attaches the identifiers (IP addresses) of the relay nodes to the packet. Upon reception of a packet, a node rebroadcasts the packet only if itself is included in the relay node list.

Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks 209

Every node maintains a distance factor, mobility factor and signal strength factor for each neighbor. These factors are updated upon reception of a hello message. Before sending a data packet, each node evaluates one-hop neighbors by using fuzzy logic to combine these factors. Based on the evaluation result, the nodes which have high evaluation values are selected as

The sender node specifies relay nodes. It is important to ensure selected relay nodes reaching all intended receivers while minimizing the number of rebroadcasts. To solve this issue, the concept of "broadcast zone" is introduced. In the protocol, a sender node selects one relay

A sender node first groups neighbor vehicles according to [road\_no, sender\_pos, direction]. As shown in Fig. 3, "road\_no" denotes the road number, "sender\_pos" denotes the sender position and "direction" can be "outbound" or "inbound." We call a triad [road\_no, sender\_pos, direction] a "broadcast zone". For example, the triad [1, (*x*, *y*, *z*), outbound] shows the area which is on the road No.1 and in the "outbound" direction of position (*x*, *y*, *z*).

We note that "outbound" and "inbound" are predefined for each road. For a loop-free road, since the start point and end point can be defined, we define the direction from the start point to the end point as "outbound," and define the direction from the end point to the start point as "inbound." For a loop road, we define the clockwise direction as "outbound" and the counter-clockwise direction as "inbound." As shown in Fig. 3, for road No.1, the direction from A to B is the outbound direction, and the direction from B to A is the inbound direction. In here, "outbound" and "inbound" depend on the position of the vehicles but be independent to the driving directions of the vehicles. We say V1 is at the outbound direction of node V2.

Before broadcasting a data message, the source node specifies the intended area as a list of broadcast zones. The sender node selects one relay node in each of the specified broadcast zones. In the example in Fig. 3, to disseminate information in all directions, node S has to

In a large scale network, we do not need to let a data message traverse through the whole network. In this case we can specify a border for each broadcast zone by specifying the most distant (from the sender node) position of the intended area. Another way is to define a life time for each message by specifying the hop count or TTL (Time To Live). In this section, without loss of generality, we consider all nodes in the network as the intended receivers.

Fig. 2. Multi-hop broadcast by using relay nodes.

**4.2 Broadcast zone and the number of relay nodes**

In contrast, V2 is at the inbound direction of node V1.

node from each broadcast zone.

select 4 relay nodes.

relay nodes.

too expensive for practical application. Fortunately, fuzzy logic can handle imprecise and uncertain information. Therefore, we use a fuzzy logic based method to identify those relay nodes that will give the best results.

Fig. 1. Using fuzzy logic to consider multiple metrics jointly.

In fuzzy set theory (Klir et al. (1997)), elements have degrees of membership. Fuzzy set theory represents incomplete or imprecise information by defining set membership as a possibility distribution. Based on fuzzy set theory, fuzzy logic deals with the concept of approximate rather than precise factors. For example, we can define a person's height as being 0.5 "high" and 0.5 "low", rather than "completely high" or "completely low". Fuzzy logic has been broadly used for industrial communities due to its efficient handling of approximate reasoning which is similar to human reasoning. In contrast to numerical values in mathematics, fuzzy logic uses non-numeric linguistic variables to express the facts. Fuzzy logic uses fuzzy membership functions to represent the degrees of a numerical value belonging to linguistic variables.

Typically, a fuzzy logic based system consists of three steps: input, process and output steps. In the input step, numerical values are converted to linguistic variables. The process step collects fuzzy rules which are defined in the form of IF-THEN statements and applies the rules to get the result in a linguistic format. The output step converts the linguistic result into a numerical value.

A fuzzy logic based system is flexible because the system can satisfy different requirements by tuning the fuzzy membership function and fuzzy rules. A flexible design is very important for vehicular ad hoc networks due to the variance of channel status and vehicle movement for different road conditions.

#### **4. A multi-hop broadcast protocol based on fuzzy logic**

In this section, we present an approach which uses a fuzzy logic to enhance multi-hop broadcast in vehicular ad hoc networks.

#### **4.1 Protocol design**

The protocol uses a sender-oriented approach. As shown in Fig. 2. In order to reduce rebroadcast redundancy in high-density networks, the protocol uses only a subset of nodes in the network to relay broadcast packets. We assume every node knows its own position which can be acquired from GPS like positioning services. Vehicles exchange information through hello messages. Every vehicle places its own position information to hello messages and therefore vehicles know positions of their neighbors. A neighbor node is removed from the neighbor list if a node fails to receive any hello message from the neighbor node in 3 times 6 Will-be-set-by-IN-TECH

too expensive for practical application. Fortunately, fuzzy logic can handle imprecise and uncertain information. Therefore, we use a fuzzy logic based method to identify those relay

In fuzzy set theory (Klir et al. (1997)), elements have degrees of membership. Fuzzy set theory represents incomplete or imprecise information by defining set membership as a possibility distribution. Based on fuzzy set theory, fuzzy logic deals with the concept of approximate rather than precise factors. For example, we can define a person's height as being 0.5 "high" and 0.5 "low", rather than "completely high" or "completely low". Fuzzy logic has been broadly used for industrial communities due to its efficient handling of approximate reasoning which is similar to human reasoning. In contrast to numerical values in mathematics, fuzzy logic uses non-numeric linguistic variables to express the facts. Fuzzy logic uses fuzzy membership functions to represent the degrees of a numerical value belonging to linguistic variables.

Typically, a fuzzy logic based system consists of three steps: input, process and output steps. In the input step, numerical values are converted to linguistic variables. The process step collects fuzzy rules which are defined in the form of IF-THEN statements and applies the rules to get the result in a linguistic format. The output step converts the linguistic result into

A fuzzy logic based system is flexible because the system can satisfy different requirements by tuning the fuzzy membership function and fuzzy rules. A flexible design is very important for vehicular ad hoc networks due to the variance of channel status and vehicle movement for

In this section, we present an approach which uses a fuzzy logic to enhance multi-hop

The protocol uses a sender-oriented approach. As shown in Fig. 2. In order to reduce rebroadcast redundancy in high-density networks, the protocol uses only a subset of nodes in the network to relay broadcast packets. We assume every node knows its own position which can be acquired from GPS like positioning services. Vehicles exchange information through hello messages. Every vehicle places its own position information to hello messages and therefore vehicles know positions of their neighbors. A neighbor node is removed from the neighbor list if a node fails to receive any hello message from the neighbor node in 3 times

nodes that will give the best results.

a numerical value.

different road conditions.

**4.1 Protocol design**

broadcast in vehicular ad hoc networks.

Fig. 1. Using fuzzy logic to consider multiple metrics jointly.

**4. A multi-hop broadcast protocol based on fuzzy logic**

the hello interval. The hello interval is set to 1 second. Before broadcasting a packet, a sender node attaches the identifiers (IP addresses) of the relay nodes to the packet. Upon reception of a packet, a node rebroadcasts the packet only if itself is included in the relay node list.

Fig. 2. Multi-hop broadcast by using relay nodes.

Every node maintains a distance factor, mobility factor and signal strength factor for each neighbor. These factors are updated upon reception of a hello message. Before sending a data packet, each node evaluates one-hop neighbors by using fuzzy logic to combine these factors. Based on the evaluation result, the nodes which have high evaluation values are selected as relay nodes.

#### **4.2 Broadcast zone and the number of relay nodes**

The sender node specifies relay nodes. It is important to ensure selected relay nodes reaching all intended receivers while minimizing the number of rebroadcasts. To solve this issue, the concept of "broadcast zone" is introduced. In the protocol, a sender node selects one relay node from each broadcast zone.

A sender node first groups neighbor vehicles according to [road\_no, sender\_pos, direction]. As shown in Fig. 3, "road\_no" denotes the road number, "sender\_pos" denotes the sender position and "direction" can be "outbound" or "inbound." We call a triad [road\_no, sender\_pos, direction] a "broadcast zone". For example, the triad [1, (*x*, *y*, *z*), outbound] shows the area which is on the road No.1 and in the "outbound" direction of position (*x*, *y*, *z*).

We note that "outbound" and "inbound" are predefined for each road. For a loop-free road, since the start point and end point can be defined, we define the direction from the start point to the end point as "outbound," and define the direction from the end point to the start point as "inbound." For a loop road, we define the clockwise direction as "outbound" and the counter-clockwise direction as "inbound." As shown in Fig. 3, for road No.1, the direction from A to B is the outbound direction, and the direction from B to A is the inbound direction. In here, "outbound" and "inbound" depend on the position of the vehicles but be independent to the driving directions of the vehicles. We say V1 is at the outbound direction of node V2. In contrast, V2 is at the inbound direction of node V1.

Before broadcasting a data message, the source node specifies the intended area as a list of broadcast zones. The sender node selects one relay node in each of the specified broadcast zones. In the example in Fig. 3, to disseminate information in all directions, node S has to select 4 relay nodes.

In a large scale network, we do not need to let a data message traverse through the whole network. In this case we can specify a border for each broadcast zone by specifying the most distant (from the sender node) position of the intended area. Another way is to define a life time for each message by specifying the hop count or TTL (Time To Live). In this section, without loss of generality, we consider all nodes in the network as the intended receivers.

to smooth out short-term errors. The value of *α* is set to 0.7 based on out experimental results.

Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks 211

*<sup>R</sup>* ). (9)

*RxPr* ). (10)

*MF*(*X*) <sup>←</sup> (<sup>1</sup> <sup>−</sup> *<sup>α</sup>*) <sup>×</sup> *MF*(*X*) + *<sup>α</sup>* <sup>×</sup> (<sup>1</sup> <sup>−</sup> <sup>|</sup>*di*(*X*) <sup>−</sup> *di*−1(*X*)<sup>|</sup>

As shown in Eq. (9), the lower the relative movement, the larger is the mobility factor. Since each neighbor is evaluated periodically (upon reception of a hello message), a large mobility factor is required to ensure a specified relay node is still in the transmission range of the sender

Upon reception of a hello message from a neighbor *X*, a node calculates a Received Signal Strength Indication Factor (RSSIF) as Eq. (10). In Eq. (10), RxPr denotes the received signal power, and RXThresh is the reception threshold. RSSIF indicates the average signal strength

*RSSIF*(*X*) <sup>←</sup> (<sup>1</sup> <sup>−</sup> *<sup>α</sup>*) <sup>×</sup> *RSSIF*(*X*) + *<sup>α</sup>* <sup>×</sup> (<sup>1</sup> <sup>−</sup> *RXThresh*

Eq. (10) calculates the average signal strength from a neighbor node. In here, we use the RSSI factor to estimate the received signal strength at the neighbor node. A high RSSIF factor can ensure the packet reception at the neighbor node when the neighbor node is selected as a relay

As mentioned above, each node evaluates its neighbors in term of distance, mobility and signal strength by exchanging hello messages. When there is a need to send a packet, a node employs the fuzzy logic to calculate an average relay fitness value for each neighbor based on the neighbor's distance, mobility and signal strength. The node then selects a relay node for

For each broadcast zone, a sender node selects the node that has maximal fitness value to relay the packet. The calculation steps for the relay fitness value for each neighbor are as follows. • **Fuzzification** Use predefined linguistic variables and membership functions to convert the

• **Mapping and combination of IF/THEN rules** Map the fuzzy values to predefined IF/THEN rules and combine the rules to get the rank of the neighbor as a fuzzy output

• **Defuzzification** Use predefined output membership function and defuzzification method

"Fuzzification" is the process of converting a numerical value to a fuzzy value using a predefined fuzzy membership function. The fuzzy membership function of distance factor is defined as Fig. 4. The linguistic variables defined for the distance factor are {Large, Medium,

distance factor, mobility factor and RSSI factor to corresponding fuzzy values.

to convert the fuzzy output value to a numerical value.

MF is initialized to 0.

**4.3.3 Signal strength**

node.

**4.4.1 Procedure**

each broadcast zone.

value.

**4.4.2 Fuzzification**

node when a data packet is sent at the sender node.

of the neighbor node. Here RSSIF is initialized to 0.

**4.4 Relay node selection based on fuzzy logic**

Fig. 3. A street road topology.

#### **4.3 Neighborhood status update using hello messages**

In the protocol, upon reception of a hello message from a neighbor, a node evaluates the neighbor according to the inter-vehicle distance, mobility and signal strength respectively. In this way, through exchanging hello messages, each node maintains an evaluation result for each neighbor. When selecting a relay node, these evaluation results are used.

#### **4.3.1 Distance**

Upon reception of a hello message from a neighbor *X*, a node calculates a Distance Factor (DF) as Eq. (8). In Eq. (8), *d*(*X*) is the distance between the current node and node *X*. *R* is the average transmission range. Here we assume every node has the same transmission power and the transmission power is constant.

$$DF(X) = \begin{cases} \frac{d(X)}{\mathbb{R}}, & d(X) < = \mathbb{R} \\ 1, & d(X) > \mathbb{R} \end{cases} \tag{8}$$

Eq. (8) gives a higher value for a node which has larger distance from the sender node. When the Distance Factor is large, a message can reach the destination region with a small number of rebroadcasts. Therefore, a larger distance factor is desirable to provide a high efficiency.

#### **4.3.2 Mobility**

Upon reception of a hello message from a neighbor *X*, a node calculates a Mobility Factor (MF) as Eq. (9). MF indicates the mobility level of the neighbor node. Here, *di*(*X*) is the distance between the current node and the neighbor node at time *i*. *α* is a smooth factor which is used to smooth out short-term errors. The value of *α* is set to 0.7 based on out experimental results. MF is initialized to 0.

$$MF(X) \leftarrow (1 - a) \times MF(X) + a \times (1 - \frac{|d\_i(X) - d\_{i-1}(X)|}{R}).\tag{9}$$

As shown in Eq. (9), the lower the relative movement, the larger is the mobility factor. Since each neighbor is evaluated periodically (upon reception of a hello message), a large mobility factor is required to ensure a specified relay node is still in the transmission range of the sender node when a data packet is sent at the sender node.

#### **4.3.3 Signal strength**

8 Will-be-set-by-IN-TECH

In the protocol, upon reception of a hello message from a neighbor, a node evaluates the neighbor according to the inter-vehicle distance, mobility and signal strength respectively. In this way, through exchanging hello messages, each node maintains an evaluation result for

Upon reception of a hello message from a neighbor *X*, a node calculates a Distance Factor (DF) as Eq. (8). In Eq. (8), *d*(*X*) is the distance between the current node and node *X*. *R* is the average transmission range. Here we assume every node has the same transmission power

*d*(*X*)

Eq. (8) gives a higher value for a node which has larger distance from the sender node. When the Distance Factor is large, a message can reach the destination region with a small number of rebroadcasts. Therefore, a larger distance factor is desirable to provide a high efficiency.

Upon reception of a hello message from a neighbor *X*, a node calculates a Mobility Factor (MF) as Eq. (9). MF indicates the mobility level of the neighbor node. Here, *di*(*X*) is the distance between the current node and the neighbor node at time *i*. *α* is a smooth factor which is used

<sup>R</sup> , *d*(*X*) <= R

1, *<sup>d</sup>*(*X*) <sup>&</sup>gt; <sup>R</sup> (8)

each neighbor. When selecting a relay node, these evaluation results are used.

*DF*(*X*) =

Fig. 3. A street road topology.

and the transmission power is constant.

**4.3.1 Distance**

**4.3.2 Mobility**

**4.3 Neighborhood status update using hello messages**

Upon reception of a hello message from a neighbor *X*, a node calculates a Received Signal Strength Indication Factor (RSSIF) as Eq. (10). In Eq. (10), RxPr denotes the received signal power, and RXThresh is the reception threshold. RSSIF indicates the average signal strength of the neighbor node. Here RSSIF is initialized to 0.

$$RSSIF(X) \leftarrow (1 - \mathfrak{a}) \times RSSIF(X) + \mathfrak{a} \times (1 - \frac{\mathbb{R}XThresh}{RxPr}).\tag{10}$$

Eq. (10) calculates the average signal strength from a neighbor node. In here, we use the RSSI factor to estimate the received signal strength at the neighbor node. A high RSSIF factor can ensure the packet reception at the neighbor node when the neighbor node is selected as a relay node.

#### **4.4 Relay node selection based on fuzzy logic**

#### **4.4.1 Procedure**

As mentioned above, each node evaluates its neighbors in term of distance, mobility and signal strength by exchanging hello messages. When there is a need to send a packet, a node employs the fuzzy logic to calculate an average relay fitness value for each neighbor based on the neighbor's distance, mobility and signal strength. The node then selects a relay node for each broadcast zone.

For each broadcast zone, a sender node selects the node that has maximal fitness value to relay the packet. The calculation steps for the relay fitness value for each neighbor are as follows.


#### **4.4.2 Fuzzification**

"Fuzzification" is the process of converting a numerical value to a fuzzy value using a predefined fuzzy membership function. The fuzzy membership function of distance factor is defined as Fig. 4. The linguistic variables defined for the distance factor are {Large, Medium,

**4.4.3 Rule base**

Table 1. Rule Base

*Strength* is Good **THEN** *Rank* is Perfect.

This is why the Rank of the Rule1 is Perfect.

Based on the fuzzy values of distance factor, mobility factor and RSSI factor, the sender node uses the IF/THEN rules (as defined in Table 1) to calculate the rank of the node. The linguistic variables of the rank are defined as {Perfect, Good, Acceptable, NotAcceptable, Bad, VeryBad}. In Table 1, Rule1 defines the following rule. **IF** *Distance* is Large, *Mobility* is Slow and *Signal* Distance Mobility Signal Strength Rank Rule1 Large Slow Good Perfect Rule2 Large Slow Medium Good

Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks 213

Rule3 Large Slow Bad NotAcceptable

Rule7 Large Fast Good NotAcceptable

Rule4 Large Medium Good Good Rule5 Large Medium Medium Acceptable Rule6 Large Medium Bad Bad

Rule8 Large Fast Medium Bad Rule9 Large Fast Bad VeryBad Rule10 Medium Slow Good Good Rule11 Medium Slow Medium Acceptable Rule12 Medium Slow Bad Bad Rule13 Medium Medium Good Acceptable Rule14 Medium Medium Medium NotAcceptable

Rule15 Medium Medium Bad Bad Rule16 Medium Fast Good Bad Rule17 Medium Fast Medium Bad Rule18 Medium Fast Bad VeryBad Rule19 Small Slow Good NotAcceptable

Rule20 Small Slow Medium Bad Rule21 Small Slow Bad VeryBad Rule22 Small Medium Good Bad Rule23 Small Medium Medium Bad Rule24 Small Medium Bad VeryBad Rule25 Small Fast Good VeryBad Rule26 Small Fast Medium VeryBad Rule27 Small Fast Bad VeryBad

When the distance factor is large, we can reduce the number of hops for broadcast. When the mobility is slow, the relay nodes are not likely to move out the transmission range of the sender node. A high Signal Strength can ensure a packet will be received by the relay nodes.

Compared with the Rule1, when any one of three factors (Distance, Mobility and Signal Strength) drops to the next level, we set the Rank to be "Good" (Rule2, Rule4 and Rule10). Similarly, when any two of three factors drop to the next level, we set the rank to be "Acceptable" (Rule5, Rule11 and Rule13). When any one of three factors drops to the worst level, we set the Rank to be "NotAcceptable" (Rule3, Rule7 and Rule19). The same for the

Small}. The sender node uses the membership function and the distance factor to calculate what degree the distance factor belongs to {Large, Medium, Small}. As shown in Fig. 4, when the distance factor is 0.2, we get a fuzzy value {Large:0, Medium:0.4, Small:0.6}. Fig. 5 shows

Fig. 4. Distance membership function.

the fuzzy membership function defined for the mobility factor. The sender node uses the mobility factor and this membership function to calculate what degree the mobility factor belongs to {Slow, Medium, Fast}. Fig. 6 shows the fuzzy membership function defined for the

Fig. 5. Mobility membership function.

RSSI factor. The sender node uses the RSSI factor and this membership function to calculate what degree the RSSI factor belongs to {Good, Medium, Bad}.

Fig. 6. Signal strength membership function.

#### **4.4.3 Rule base**

10 Will-be-set-by-IN-TECH

Small}. The sender node uses the membership function and the distance factor to calculate what degree the distance factor belongs to {Large, Medium, Small}. As shown in Fig. 4, when the distance factor is 0.2, we get a fuzzy value {Large:0, Medium:0.4, Small:0.6}. Fig. 5 shows

Small:0.6

Medium:0.4

Small Medium Large

0 0.2 0.4 0.6 0.8 1

Distance Factor

Fast Medium Slow

Bad Medium Good

0 0.2 0.4 0.6 0.8 1

Mobility Factor

0 0.2 0.4 0.6 0.8 1

RSSI Factor

RSSI factor. The sender node uses the RSSI factor and this membership function to calculate

the fuzzy membership function defined for the mobility factor. The sender node uses the mobility factor and this membership function to calculate what degree the mobility factor belongs to {Slow, Medium, Fast}. Fig. 6 shows the fuzzy membership function defined for the

 0 0.2 0.4 0.6 0.8 1

 0 0.2 0.4 0.6 0.8 1

 0 0.2 0.4 0.6 0.8 1

what degree the RSSI factor belongs to {Good, Medium, Bad}.

Degree

Fig. 5. Mobility membership function.

Degree

Fig. 6. Signal strength membership function.

Degree

Fig. 4. Distance membership function.

Based on the fuzzy values of distance factor, mobility factor and RSSI factor, the sender node uses the IF/THEN rules (as defined in Table 1) to calculate the rank of the node. The linguistic variables of the rank are defined as {Perfect, Good, Acceptable, NotAcceptable, Bad, VeryBad}. In Table 1, Rule1 defines the following rule. **IF** *Distance* is Large, *Mobility* is Slow and *Signal*


Table 1. Rule Base

*Strength* is Good **THEN** *Rank* is Perfect.

When the distance factor is large, we can reduce the number of hops for broadcast. When the mobility is slow, the relay nodes are not likely to move out the transmission range of the sender node. A high Signal Strength can ensure a packet will be received by the relay nodes. This is why the Rank of the Rule1 is Perfect.

Compared with the Rule1, when any one of three factors (Distance, Mobility and Signal Strength) drops to the next level, we set the Rank to be "Good" (Rule2, Rule4 and Rule10). Similarly, when any two of three factors drop to the next level, we set the rank to be "Acceptable" (Rule5, Rule11 and Rule13). When any one of three factors drops to the worst level, we set the Rank to be "NotAcceptable" (Rule3, Rule7 and Rule19). The same for the

0

gravity is calculated as

has the maximal fitness value.

**4.5 Simulation results**

Fig. 8. Output membership function and an example for *μ*(*x*).

0 0.2 0.4 0.6 0.8 1

*μ*(*x*)*xdx*

*<sup>μ</sup>*(*x*)*dx* , (11)

above, when the degree for Rank {Acceptable} is 0.2, the degree for Rank {Good} is 0.5 and the degree for Rank {Perfect} is 0.5, the result function will be as shown in Fig. 8. The center of

where *μ*(*x*) is the result function and *x* is the value of X-axis. In this protocol, the calculated COG represents the fitness of the neighbor being a relay node. For each broadcast zone, the sender node calculates a fitness value for each neighbor node and then selects the node which

Network Simulator 2 (ns-2.34) (ns-2 (2010)) was used to conduct simulations. We used a Freeway model (Bai et al. (2003)) to generate the network topology (see Table 2). We used a freeway which has two lanes in each direction. All lanes of the freeway were 2000 m in length. The maximum allowable vehicle velocity was 40m/s. We used Nakagami propagation model. Parameters of the Nakagami model are shown in Table 3. These parameters result packet delivery ratios as shown in Fig. 9. We used these parameter values because they model a

*COG* =

realistic wireless channel of vehicular ad hoc networks (Khan et al. (2009)).

Number of packets 50 packets at each source

Data rate 10 packet per second MAC IEEE 802.11 MAC (2Mbps)

Propagation model Nakagami Model

Number of nodes 100 to 600 Mobility generation Bai et al. (2003)

Packet size 512 bytes

Simulation time 150 s

Table 2. Simulation Environment

Number of sources 2

Topology Freeway scenario, 2000m, 4lanes

Number of receivers The number of all nodes in the network

Other simulation parameters were the default settings of ns-2.34. From 20s, two source nodes generated 50 packets with a rate of 10 packets per second. These two nodes (randomly

VeryBad Bad NotAcceptable Acceptable Good Perfect

Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks 215

0.2

0.4

0.6

0.8

1

case when all three factors are at the medium level (Rule 14). When two or all three factors drop to the worst level, we set the Rank to be "VeryBad" (Rule9, Rule18, Rule21, Rule24, Rule 25, Rule26 and Rule27). For other rules, we set the Rank to be "Bad" (Rule6, Rule8, Rule12, Rule15, Rule16, Rule17, Rule20, Rule22 and Rule23). In this way, we define 27 rules in total. These rules cover all possible combinations of fuzzy values in different factors.

In a rule, the IF part is called the "antecedent" and the THEN part is called the "consequent". Since there can be multiple rules applying for the same fuzzy variables, we have to combine their evaluation results. Here we use Min-Max method to match and combine the rules. In the Min-Max method, for each rule, the minimal value of antecedent is used as the final degree. When combining different rules, the maximal value of consequents is used.

For example, as shown in Fig. 7, we assume a neighbor's distance, mobility and RSSI factor belong to the corresponding linguistic variables as {Large:1, Medium:0, Small:0},{Slow:0.8, Medium:0.2, Fast:0},{Good:0.5, Medium:0.5, Bad:0} respectively. In this case, these fuzzy sets match Rule1, Rule2, Rule4 and Rule5. For Rule1, the degree for {Large} (Distance) is 1, the degree for {Slow} (Mobility) is 0.8 and the degree for {Good} (Signal Strength) is 0.5. In the Min-Max method, we take the minimal value of antecedent members and therefore the degree of the antecedent will be 0.5. Similarly, the degrees of antecedents for Rule2, Rule4 and Rule5 will be 0.5, 0.2 and 0.2 respectively. As both Rule2 and Rule4 lead to the Rank {Good}, we take the maximal value of consequents and therefore the degree of the Rank Good will be 0.5. In this way, all rules are combined to get a fuzzy result.

Fig. 7. An example for fuzzy rule evaluations.

#### **4.4.4 Defuzzification**

Defuzzification is used to produce a numeric result based on a predefined output membership function and corresponding membership degrees. Fig. 8 shows the defined output membership function. Here Center of Gravity (COG) method is used to defuzzify the fuzzy result.

As shown in Fig. 8, we cut the output membership function in a straight horizontal line according to the corresponding degree, and remove the top portion. For the example given

Fig. 8. Output membership function and an example for *μ*(*x*).

above, when the degree for Rank {Acceptable} is 0.2, the degree for Rank {Good} is 0.5 and the degree for Rank {Perfect} is 0.5, the result function will be as shown in Fig. 8. The center of gravity is calculated as

$$\text{COG} = \frac{\int \mu(\mathbf{x}) \mathbf{x} d\mathbf{x}}{\int \mu(\mathbf{x}) d\mathbf{x}},\tag{11}$$

where *μ*(*x*) is the result function and *x* is the value of X-axis. In this protocol, the calculated COG represents the fitness of the neighbor being a relay node. For each broadcast zone, the sender node calculates a fitness value for each neighbor node and then selects the node which has the maximal fitness value.

#### **4.5 Simulation results**

12 Will-be-set-by-IN-TECH

case when all three factors are at the medium level (Rule 14). When two or all three factors drop to the worst level, we set the Rank to be "VeryBad" (Rule9, Rule18, Rule21, Rule24, Rule 25, Rule26 and Rule27). For other rules, we set the Rank to be "Bad" (Rule6, Rule8, Rule12, Rule15, Rule16, Rule17, Rule20, Rule22 and Rule23). In this way, we define 27 rules in total.

In a rule, the IF part is called the "antecedent" and the THEN part is called the "consequent". Since there can be multiple rules applying for the same fuzzy variables, we have to combine their evaluation results. Here we use Min-Max method to match and combine the rules. In the Min-Max method, for each rule, the minimal value of antecedent is used as the final degree.

For example, as shown in Fig. 7, we assume a neighbor's distance, mobility and RSSI factor belong to the corresponding linguistic variables as {Large:1, Medium:0, Small:0},{Slow:0.8, Medium:0.2, Fast:0},{Good:0.5, Medium:0.5, Bad:0} respectively. In this case, these fuzzy sets match Rule1, Rule2, Rule4 and Rule5. For Rule1, the degree for {Large} (Distance) is 1, the degree for {Slow} (Mobility) is 0.8 and the degree for {Good} (Signal Strength) is 0.5. In the Min-Max method, we take the minimal value of antecedent members and therefore the degree of the antecedent will be 0.5. Similarly, the degrees of antecedents for Rule2, Rule4 and Rule5 will be 0.5, 0.2 and 0.2 respectively. As both Rule2 and Rule4 lead to the Rank {Good}, we take the maximal value of consequents and therefore the degree of the Rank Good will be 0.5. In

Defuzzification is used to produce a numeric result based on a predefined output membership function and corresponding membership degrees. Fig. 8 shows the defined output membership function. Here Center of Gravity (COG) method is used to defuzzify the fuzzy

As shown in Fig. 8, we cut the output membership function in a straight horizontal line according to the corresponding degree, and remove the top portion. For the example given

These rules cover all possible combinations of fuzzy values in different factors.

When combining different rules, the maximal value of consequents is used.

this way, all rules are combined to get a fuzzy result.

Fig. 7. An example for fuzzy rule evaluations.

**4.4.4 Defuzzification**

result.

Network Simulator 2 (ns-2.34) (ns-2 (2010)) was used to conduct simulations. We used a Freeway model (Bai et al. (2003)) to generate the network topology (see Table 2). We used a freeway which has two lanes in each direction. All lanes of the freeway were 2000 m in length. The maximum allowable vehicle velocity was 40m/s. We used Nakagami propagation model. Parameters of the Nakagami model are shown in Table 3. These parameters result packet delivery ratios as shown in Fig. 9. We used these parameter values because they model a realistic wireless channel of vehicular ad hoc networks (Khan et al. (2009)).


Table 2. Simulation Environment

Other simulation parameters were the default settings of ns-2.34. From 20s, two source nodes generated 50 packets with a rate of 10 packets per second. These two nodes (randomly

0

Fig. 11. Packet dissemination ratio for various number of nodes.

0.2

0.4

0.6

Packet dissemination ratio

0.8

1

Number of messages per data packet

ratio.

100 200 300 400 500 600

Flooding Weighted p-persistence

EMPR without retransmission

Fig. 10. Number of broadcasts per data packet for various number of nodes.

MPR

Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks 217

Fuzzy

Number of nodes

100 200 300 400 500 600

Flooding Weighted p-persistence

EMPR without retransmission

MPR

Fuzzy

Number of nodes

The Weighted *p*-persistence scheme works better than the flooding by reducing the number of broadcasts. However, since a probabilistic method is used, the number of broadcasts also increases as the node density increases, leading to a drop in performance. In the MPR Broadcast, although the number of broadcasts can be efficiently reduced, we observe a poor dissemination ratio. This is because a sender node usually selects the farthest node. However, in a fading channel, the furthest node always fails to receive the broadcast packet. In MPR, since the node mobility is not considered in the relay node selection, a packet loss also occurs at the selected relay node due to the vehicle movement. The EMPR Broadcast performs better than the MPR Broadcast because it considers node mobility in the relay node selection. In the EMPR Broadcast, a sender node selects a relay node which has a low relative mobility

to broadcast at the same time and this introduces collisions and a drop in packet dissemination


Table 3. Parameters of Nakagami Model

Fig. 9. Packet reception probability for various distances.

selected) were neighbors and being close to each other. This is to simulate a condition of two collided vehicles send data messages at the same time. Simulation time was 150s. We launched simulations with 50 different vehicle deployments and different vehicle movements, and analyzed the average value.

The protocol (Fuzzy) was compared with Flooding, Weighted *p*-persistence (Wisitpongphan & Tonguz (2007)), MPR Broadcast (Qayyum et al. (2002)) and EMPR Broadcast (Wu et al. (2010)). We did not use retransmission in all these protocols.

#### **4.5.1 Number of broadcasts**

Fig. 10 shows the number of broadcasts per data packet for various number of nodes. Flooding generates too many redundant broadcasts in a high density network. As a result, many packets are lost due to packet collisions.

Since the Weighted *p*-persistence uses a probabilistic broadcast method to reduce the redundant rebroadcast, the Weighted *p*-persistence performs better than the flooding. However, the number of broadcasts also increases linearly with the increase of node density. Therefore, redundant rebroadcasts cannot be eliminated entirely. In the MPR Broadcast, EMPR Broadcast and the Fuzzy protocol, only the nodes which have been selected as relay nodes, rebroadcast the packets. Therefore, the redundant broadcast can be reduced efficiently.

#### **4.5.2 Packet dissemination ratio**

Fig. 11 shows packet dissemination ratio for various number of nodes. In flooding, as the number of nodes increases, the dissemination ratio decreases. This is because many nodes try 14 Will-be-set-by-IN-TECH

gamma0\_ gamma1\_ gamma2\_ d0\_gamma\_ d1\_gamma\_

0 100 200 300 400 500 600 700

Distance

selected) were neighbors and being close to each other. This is to simulate a condition of two collided vehicles send data messages at the same time. Simulation time was 150s. We launched simulations with 50 different vehicle deployments and different vehicle movements,

The protocol (Fuzzy) was compared with Flooding, Weighted *p*-persistence (Wisitpongphan & Tonguz (2007)), MPR Broadcast (Qayyum et al. (2002)) and EMPR

Fig. 10 shows the number of broadcasts per data packet for various number of nodes. Flooding generates too many redundant broadcasts in a high density network. As a result, many

Since the Weighted *p*-persistence uses a probabilistic broadcast method to reduce the redundant rebroadcast, the Weighted *p*-persistence performs better than the flooding. However, the number of broadcasts also increases linearly with the increase of node density. Therefore, redundant rebroadcasts cannot be eliminated entirely. In the MPR Broadcast, EMPR Broadcast and the Fuzzy protocol, only the nodes which have been selected as relay nodes, rebroadcast the packets. Therefore, the redundant broadcast can be reduced efficiently.

Fig. 11 shows packet dissemination ratio for various number of nodes. In flooding, as the number of nodes increases, the dissemination ratio decreases. This is because many nodes try

Broadcast (Wu et al. (2010)). We did not use retransmission in all these protocols.

1.9 3.8 3.8 200 500 m0\_ m1\_ m2\_ d0\_m\_ d1\_m\_ 1.5 0.75 0.75 80 200

Table 3. Parameters of Nakagami Model

0

Fig. 9. Packet reception probability for various distances.

0.2

0.4

0.6

Packet Reception Probability

and analyzed the average value.

**4.5.1 Number of broadcasts**

packets are lost due to packet collisions.

**4.5.2 Packet dissemination ratio**

0.8

1

Fig. 10. Number of broadcasts per data packet for various number of nodes.

to broadcast at the same time and this introduces collisions and a drop in packet dissemination ratio.

Fig. 11. Packet dissemination ratio for various number of nodes.

The Weighted *p*-persistence scheme works better than the flooding by reducing the number of broadcasts. However, since a probabilistic method is used, the number of broadcasts also increases as the node density increases, leading to a drop in performance. In the MPR Broadcast, although the number of broadcasts can be efficiently reduced, we observe a poor dissemination ratio. This is because a sender node usually selects the farthest node. However, in a fading channel, the furthest node always fails to receive the broadcast packet. In MPR, since the node mobility is not considered in the relay node selection, a packet loss also occurs at the selected relay node due to the vehicle movement. The EMPR Broadcast performs better than the MPR Broadcast because it considers node mobility in the relay node selection. In the EMPR Broadcast, a sender node selects a relay node which has a low relative mobility

0

Fig. 13. End-to-end delay for various number of nodes.

does show a low delay even when the network density is high.

100 200 300 400 500 600

Flooding Weighted p-persistence

EMPR without retransmission

MPR

Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks 219

Fuzzy

Number of nodes

MPR shows the lowest delay. This is because MPR chooses the farthest node as a relay node.

EMPR Broadcast and the Fuzzy protocol show comparable delays. Although the selected relay nodes are usually not the farthest possible nodes, the Fuzzy protocol shows lower end-to-end delays. This is because the Fuzzy protocol reduces the contention time at each node by reducing the number of rebroadcasts. The Fuzzy protocol shows an increase of the end-to-end delay with the increase of the number of nodes. This is because with the increase of node density, the number of hello messages increases, resulting in a slight increase of MAC layer contention time at each node. However, this is acceptable because the Fuzzy protocol

Efficient and reliable relay node selection is important for providing multi-hop broadcast services in vehicular ad hoc networks. Due to the network dynamics of vehicular ad hoc networks, the optimal mathematical model of the relay node selection problem is difficult to derive. As a solution, in this chapter, we presented a fuzzy logic protocol to enhance the multi-hop broadcast in vehicular ad hoc networks. By employing the fuzzy logic into the relay node selection, the protocol considers the inter-vehicle distance, node mobility and signal strength jointly. As a result, a high level of reliability and efficiency are provided. We used computer simulations to evaluate the protocol's performance. The simulation results confirmed that the Fuzzy protocol offers a significant performance advantage over existing alternatives by selecting better relay nodes. The fuzzy logic based approach is easy to implement and can be configured to any scenario by tuning the fuzzy membership

This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) #23700072.

The low delay of MPR is also because many data messages are lost at the relay node.

0.2

0.4

0.6

0.8

Delay (s)

**5. Conclusions**

parameters.

**6. Acknowledgement**

1

1.2

1.4

and large additional coverage. As the number of nodes increases, the choices increase and therefore the performance of the EMPR Broadcast improves slightly.

The Fuzzy protocol evaluates relay fitness values of relay nodes considering inter-vehicle distance, node mobility and received signal strength. We use Fig. 12 to show the distribution of relay fitness values for various distances and relative velocities. In here, the received signal power on a certain distance is calculated by averaging received signal powers of 10,000 packets in the same distance.

Fig. 12. Relay fitness for various distances and relative velocities.

By jointly considering inter-vehicle distance, node mobility and signal strength, the Fuzzy protocol can deal with node mobility and fading while providing large progress on the dissemination direction. As a result, the Fuzzy protocol provides better packet dissemination ratio (above 94%) than other protocols. The very small number of packet losses are because of the packet collisions. It is possible to get a higher packet reception ratio if we use a retransmission mechanism. However, this is beyond the scope of this work.

#### **4.5.3 End-to-end delay**

Fig. 13 shows end-to-end delay for various number of nodes. In the end-to-end delay calculation, we only count the successfully delivered packets. In Flooding, as the node density increases, the delay increases drastically. This is because of the increase of MAC layer contention time with the increase of the number of rebroadcasts. Another reason is the effect of packet losses. When the node density is high, the redundant broadcasts introduce many collisions and consequently the nodes that provide larger progress on distance lose the data packets. As a result, the packets are delayed because they are delivered through sub-optimal paths (longer paths).

In Weighted *p*-persistence, the end-to-end delay also increases with the increase of the node density because Weighted *p*-persistence cannot eliminate redundant broadcasts completely.

Fig. 13. End-to-end delay for various number of nodes.

MPR shows the lowest delay. This is because MPR chooses the farthest node as a relay node. The low delay of MPR is also because many data messages are lost at the relay node.

EMPR Broadcast and the Fuzzy protocol show comparable delays. Although the selected relay nodes are usually not the farthest possible nodes, the Fuzzy protocol shows lower end-to-end delays. This is because the Fuzzy protocol reduces the contention time at each node by reducing the number of rebroadcasts. The Fuzzy protocol shows an increase of the end-to-end delay with the increase of the number of nodes. This is because with the increase of node density, the number of hello messages increases, resulting in a slight increase of MAC layer contention time at each node. However, this is acceptable because the Fuzzy protocol does show a low delay even when the network density is high.

#### **5. Conclusions**

16 Will-be-set-by-IN-TECH

and large additional coverage. As the number of nodes increases, the choices increase and

The Fuzzy protocol evaluates relay fitness values of relay nodes considering inter-vehicle distance, node mobility and received signal strength. We use Fig. 12 to show the distribution of relay fitness values for various distances and relative velocities. In here, the received signal power on a certain distance is calculated by averaging received signal powers of 10,000

500 0 5 10 15 20 25 30 35<sup>40</sup>

By jointly considering inter-vehicle distance, node mobility and signal strength, the Fuzzy protocol can deal with node mobility and fading while providing large progress on the dissemination direction. As a result, the Fuzzy protocol provides better packet dissemination ratio (above 94%) than other protocols. The very small number of packet losses are because of the packet collisions. It is possible to get a higher packet reception ratio if we use a

Fig. 13 shows end-to-end delay for various number of nodes. In the end-to-end delay calculation, we only count the successfully delivered packets. In Flooding, as the node density increases, the delay increases drastically. This is because of the increase of MAC layer contention time with the increase of the number of rebroadcasts. Another reason is the effect of packet losses. When the node density is high, the redundant broadcasts introduce many collisions and consequently the nodes that provide larger progress on distance lose the data packets. As a result, the packets are delayed because they are delivered through sub-optimal

In Weighted *p*-persistence, the end-to-end delay also increases with the increase of the node density because Weighted *p*-persistence cannot eliminate redundant broadcasts completely.

Velocity (m/s)

therefore the performance of the EMPR Broadcast improves slightly.

packets in the same distance.

 0 100 200 300 400

Fitness

Fig. 12. Relay fitness for various distances and relative velocities.

retransmission mechanism. However, this is beyond the scope of this work.

 0 0.2 0.4 0.6 0.8 1

Distance (m)

**4.5.3 End-to-end delay**

paths (longer paths).

Efficient and reliable relay node selection is important for providing multi-hop broadcast services in vehicular ad hoc networks. Due to the network dynamics of vehicular ad hoc networks, the optimal mathematical model of the relay node selection problem is difficult to derive. As a solution, in this chapter, we presented a fuzzy logic protocol to enhance the multi-hop broadcast in vehicular ad hoc networks. By employing the fuzzy logic into the relay node selection, the protocol considers the inter-vehicle distance, node mobility and signal strength jointly. As a result, a high level of reliability and efficiency are provided. We used computer simulations to evaluate the protocol's performance. The simulation results confirmed that the Fuzzy protocol offers a significant performance advantage over existing alternatives by selecting better relay nodes. The fuzzy logic based approach is easy to implement and can be configured to any scenario by tuning the fuzzy membership parameters.

#### **6. Acknowledgement**

This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) #23700072.

**0**

**11**

*Spain*

**Fuzzy Logic Applied to Decision Making in**

This chapter presents a real application of fuzzy logic applied to decision making in Wireless Sensor Networks (WSNs). These networks are composed by a large number of sensor devices that communicate with each other via wireless channel, with limitations of energy and computing capabilities. The efficient and robust realization of such large, highly dynamic and complex networking environments is a challenging algorithmic and technological task. Networking is important because it provides the glue that allows individual nodes to collaborate. Radio communication is the major consumer of energy in small sensor nodes. Thus, the optimization of networking protocols can greatly extend the lifetime of the sensor

Organizing a network, composed in many cases by a high number of low-resourced nodes, is a difficult task since the algorithms and methods have to save as much energy as possible while offering good performance. Power saving has been the main driving force behind the

The design and implementation of routing schemes that are able to effectively and efficiently support information exchange and processing in WSNs is a complex task. Developers must consider a number of theoretical issues and practical limitations such as energy and

Self-organization algorithms also provide network load balance to extend network lifetime, improving efficiency, and reducing data loss. Another feature to bear in mind is network monitoring, necessary to control topology changes and the addition or elimination of nodes

We propose the use of fuzzy logic in the decision-making processes of the AODV routing protocol, in order to select the best nodes to be part of the routes. In this chapter, fuzzy logic improve the selection of routing metrics. It details parameter selection and definition, and fuzzy-rule set design. Finally, we show a complete series of results, where our intelligent proposal is compared to AODV, the routing protocol for mesh networks used by the ZigBee standard, and with AODV-ETX, an interesting metric commonly used in wireless networks. From results obtained we can afford that AODV-FL (AODV with Fuzzy Logic) consumes less energy, since it sends less discovery messages resulting in fewer collisions; the number of hops for the routes created is lower with respect to AODV and the end-to-end delay is also reduced.

development of several protocols that have recently been introduced.

**1. Introduction**

network as a whole.

computation restrictions.

in the network.

**Wireless Sensor Networks**

Antonio M. Ortiz and Teresa Olivares *Albacete Research Institute of Informatics*

#### **7. References**


### **Fuzzy Logic Applied to Decision Making in Wireless Sensor Networks**

Antonio M. Ortiz and Teresa Olivares *Albacete Research Institute of Informatics Spain*

#### **1. Introduction**

18 Will-be-set-by-IN-TECH

220 Fuzzy Logic – Emerging Technologies and Applications

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Suriyapaiboonwattana, K.; Pornavalai, C. & Chakraborty, G. (2009). An adaptive alert message

Slavik, M. & Mahgoub I. (2010). Stochastic Broadcast for VANET. *Proceedings of IEEE Consumer*

Mylonas, Y.; Lestas M. & Pitsillides A. (2008). Speed adaptive probabilistic flooding in

Qayyum, A.; Viennot L. & Laouiti A. (2002). Multipoint Relaying for Flooding Broadcast

Djedid, L.O.; Lagraa N.; Yagoubi M. & Tahari K. (2008). Adaptation of the MCDS broadcasting

Sahoo, J.; Wu E.H.K.; Sahu P.K. & Gerla M. (2009). BPAB: Binary Partition Assisted Emergency

*Computer Communications and Networks*, San Francisco, USA, pp.1–6, 2009. Klir, G.J.; Clair, U.S. & Bo, Y. (1997). *Fuzzy set theory: foundations and applications*, Prentice-Hall

Bai, F.; Sadagopan N. & Helmy A. (2003). Important: A Framework to Systematically Analyze

Wu, C.; Ohzahata S. & Kato T. (2010). Fuzzy logic based multi-hop broadcast for high-density

Khan, A.; Sadhu S. & Yeleswarapu M. (2009). A comparative analysis of DSRC and 802.11

Vehicular Ad Hoc Networks. *IEEE Wireless Communications*, Vol. 14, No.6, pp.84–94,

dissemination protocol for VANET to improve road safety. *Proceedings of IEEE Intl.*

cooperative emergency warning. *Proceedings of 4th Annual Intl. Conf. on Wireless*

Messages in Mobile Wireless Networks. *Proceedings of 35th Annual Hawaii Intl. Conf.*

protocol to VANET safety applications. *Proceedings of Intl. Conf. on Innovations in*

Broadcast Protocol For Vehicular Ad Hoc Networks. *Proceedings of 18th Intl. Conf. on*

The Impact of Mobility on Performance of Routing Protocols for Adhoc Networks. *Proceedings of 22nd Annual Joint Conf. of the IEEE Computer and Communications*

vehicular ad hoc networks. *Proceedings of IEEE Vehicular Networking Conference*, New

over Vehicular Ad hoc Networks. *Project Report*, Department of Computer Science,

The Network Simulator - ns-2. *http://www.isi.edu/nsnam/ns/*, Accessed on June 23. 2010. Wisitpongphan, N. & Tonguz, K.O. (2010). Broadcast Storm Mitigation Techniques in

*Conf. on Fuzzy Systems*, Jeju Island, Korea, pp.1639–1644, 2009.

*Communications and Networking Conference*, pp.1–5, 2010.

*on System Sciences*, Big Island, Hawaii, pp.3866–3875, 2002.

*Internet*, Maui, Hawaii, pp.1–7, 2008.

*Information Technology*, pp.534–538, 2008.

*Societies*, San Francisco, USA, pp.825–835, 2003.

University of Californai, Santa Barbara, pp.1–8, 2009.

Inc., ISBN:978-0133410587.

Jersey, USA, pp.17–24, 2010.

**7. References**

18, Oct., 2003.

2007.

This chapter presents a real application of fuzzy logic applied to decision making in Wireless Sensor Networks (WSNs). These networks are composed by a large number of sensor devices that communicate with each other via wireless channel, with limitations of energy and computing capabilities. The efficient and robust realization of such large, highly dynamic and complex networking environments is a challenging algorithmic and technological task.

Networking is important because it provides the glue that allows individual nodes to collaborate. Radio communication is the major consumer of energy in small sensor nodes. Thus, the optimization of networking protocols can greatly extend the lifetime of the sensor network as a whole.

Organizing a network, composed in many cases by a high number of low-resourced nodes, is a difficult task since the algorithms and methods have to save as much energy as possible while offering good performance. Power saving has been the main driving force behind the development of several protocols that have recently been introduced.

The design and implementation of routing schemes that are able to effectively and efficiently support information exchange and processing in WSNs is a complex task. Developers must consider a number of theoretical issues and practical limitations such as energy and computation restrictions.

Self-organization algorithms also provide network load balance to extend network lifetime, improving efficiency, and reducing data loss. Another feature to bear in mind is network monitoring, necessary to control topology changes and the addition or elimination of nodes in the network.

We propose the use of fuzzy logic in the decision-making processes of the AODV routing protocol, in order to select the best nodes to be part of the routes. In this chapter, fuzzy logic improve the selection of routing metrics. It details parameter selection and definition, and fuzzy-rule set design. Finally, we show a complete series of results, where our intelligent proposal is compared to AODV, the routing protocol for mesh networks used by the ZigBee standard, and with AODV-ETX, an interesting metric commonly used in wireless networks.

From results obtained we can afford that AODV-FL (AODV with Fuzzy Logic) consumes less energy, since it sends less discovery messages resulting in fewer collisions; the number of hops for the routes created is lower with respect to AODV and the end-to-end delay is also reduced.

Fig. 1. Maxfor Tip node (*Maxfor Technology INC. http://http://www.maxfor.co.kr*, 2011)

possible, the network lifetime.

monitoring and prevention, etc.

*http://www.wisevine.info/*, 2011).

connection.

**2.2 Applications**

Since the wireless interface is the component with highest energy consumption, communication protocols should be energy efficient, with the aim of increasing, as much as

Fuzzy Logic Applied to Decision Making in Wireless Sensor Networks 223

Data collected by nodes are usually sent to a central node or Base Station, that have higher computation capabilities that sensor nodes, higher storage capacity and used to be connected to a wired network in order to be able to access network data by using a common Internet

There is a wide variety of sensors that fulfil the requirements of any application, such as

There exists a wide range of applications for wireless sensor networks. The variety in parameters that can be read by sensors makes the number of applications to grow every day. The application range includes industrial monitoring, building and home automation, medicine, environmental monitoring, urban sensor networks or energy management among others (Vasseur, 2010). These networks can also be used for security, military defense, disaster

Applications based on sensor networks are usually focused on monitoring parameters along time, in zones where it is not possible to deploy a wired network. This parameter monitoring collects data by using wireless nodes equipped with several sensors, and the information is normally sent to a central node that gathers the information of all network nodes. Figure 2 shows a WSN node attached to a vine in the Wisevine project (*Wisevine project,*

Due to the high number of nodes that can be deployed, and its battery-based nature, nodes must be able to self-organize by themselves, in order to perform efficient and automatic

temperature, humidity, atmospheric pressure, presence, energy consumption or *CO*2.

Therefore, the use of fuzzy logic as a metric in network routing improves the performance of the overall network.

#### **2. Wireless Sensor Networks**

Wireless Sensor Networks are composed by a set of sensor nodes, it is, embedded systems that can take data from the environment such as temperature, humidity or atmospheric pressure among others, and that can communicate via wireless (Yick et al., 2008; Zhao & Guibas, 2004). Usually, data is gather in an special node, know as Base Station, central node or sink. This node is usually connected to a PC or a high capacity device. When data is taken by sensors, nodes process the information and send it to the Base Station by using diverse communication protocols.

This kind of networks can be used in any environment where continuous monitoring is necessary, and node deployment may not follow any order. Algorithms and protocols used, must be able to work autonomously, in order to efficiently satisfy application requirements.

Due to node nature and the particular applications executed in WSNs, there are several special characteristics that define this kind of networks, as well as those inherit from traditional wireless systems:


All these features make WSNs a challenging field, and several universities, enterprises and research centres are working on the design and development of effective and efficient applications and protocols for these networks.

#### **2.1 Devices**

Nodes composing WSNs are quipped with a motherboard that incorporates: micro-controller, work and secondary memory, wireless interface and input/output system. Sensors are usually plugged in the input/output system, but some recent nodes already incorporate several sensors in the motherboard (see Fig. 1).

Fig. 1. Maxfor Tip node (*Maxfor Technology INC. http://http://www.maxfor.co.kr*, 2011)

Since the wireless interface is the component with highest energy consumption, communication protocols should be energy efficient, with the aim of increasing, as much as possible, the network lifetime.

Data collected by nodes are usually sent to a central node or Base Station, that have higher computation capabilities that sensor nodes, higher storage capacity and used to be connected to a wired network in order to be able to access network data by using a common Internet connection.

There is a wide variety of sensors that fulfil the requirements of any application, such as temperature, humidity, atmospheric pressure, presence, energy consumption or *CO*2.

#### **2.2 Applications**

2 Will-be-set-by-IN-TECH

Therefore, the use of fuzzy logic as a metric in network routing improves the performance of

Wireless Sensor Networks are composed by a set of sensor nodes, it is, embedded systems that can take data from the environment such as temperature, humidity or atmospheric pressure among others, and that can communicate via wireless (Yick et al., 2008; Zhao & Guibas, 2004). Usually, data is gather in an special node, know as Base Station, central node or sink. This node is usually connected to a PC or a high capacity device. When data is taken by sensors, nodes process the information and send it to the Base Station by using diverse communication

This kind of networks can be used in any environment where continuous monitoring is necessary, and node deployment may not follow any order. Algorithms and protocols used, must be able to work autonomously, in order to efficiently satisfy application requirements. Due to node nature and the particular applications executed in WSNs, there are several special characteristics that define this kind of networks, as well as those inherit from traditional

• **Limitations:** nodes composing WSNs are small and do not permit the incorporation of powerful processors and high capacity storage devices. Furthermore, the available energy,

• **Scalability:** the large number of nodes that can be deployed to fulfil a certain task, can be much larger than traditional local-area networks, so the communication techniques for WSNs must keep its functionality and efficiency as the number of network nodes grows. • **Self-configuration:** WSNs should be able to self-configure due to manual configuration of hundreds or thousands of devices may not be possible. Moreover, the network have to self-adapt to possible changes related to the incorporation, elimination, and change of

• **Simplicity:** as a consequence of node limitations and network size, applications and

• **Specificity:** there is a big variety of parameters and available options when designing a WSN that makes designs high application dependant, and this is why most of the

All these features make WSNs a challenging field, and several universities, enterprises and research centres are working on the design and development of effective and efficient

Nodes composing WSNs are quipped with a motherboard that incorporates: micro-controller, work and secondary memory, wireless interface and input/output system. Sensors are usually plugged in the input/output system, but some recent nodes already incorporate

proposals available in the literature are focused to determined applications.

provided by batteries, limits node-operation time.

the overall network.

protocols.

wireless systems:

location of the nodes.

**2.1 Devices**

protocols must be as simple as possible.

applications and protocols for these networks.

several sensors in the motherboard (see Fig. 1).

**2. Wireless Sensor Networks**

There exists a wide range of applications for wireless sensor networks. The variety in parameters that can be read by sensors makes the number of applications to grow every day. The application range includes industrial monitoring, building and home automation, medicine, environmental monitoring, urban sensor networks or energy management among others (Vasseur, 2010). These networks can also be used for security, military defense, disaster monitoring and prevention, etc.

Applications based on sensor networks are usually focused on monitoring parameters along time, in zones where it is not possible to deploy a wired network. This parameter monitoring collects data by using wireless nodes equipped with several sensors, and the information is normally sent to a central node that gathers the information of all network nodes. Figure 2 shows a WSN node attached to a vine in the Wisevine project (*Wisevine project, http://www.wisevine.info/*, 2011).

Due to the high number of nodes that can be deployed, and its battery-based nature, nodes must be able to self-organize by themselves, in order to perform efficient and automatic

Fig. 3. ZigBee protocol stack.

**2.4 Self-organization and routing**

The ZigBee Alliance *ZigBee Specification, ZigBee Alliance* (2011) and the IEEE 802.15.4 *IEEE Standard for Part 15.4: Wireless Medium Access Control (MAC) and Physical Layer (PHY) specifications for Low-Rate Wireless Personal Area Networks (WPANS)* (2011) Task Group are leading the efforts to define a standard protocol stack for the implementation of wireless sensor networks. IEEE 802.15.4 is focused on the standardization of the MAC and physical

Fuzzy Logic Applied to Decision Making in Wireless Sensor Networks 225

The ZigBee network layer includes three different topologies, namely, tree, mesh, and cluster-based topologies. This chapter is focused on mesh topology networks for which ZigBee uses the Ad hoc On demand Distance Vector protocol (AODV), that will be detailed

The correct operation of both wired and wireless networks requires some kind of network organization. Most of networking systems follow some kind of organization, well centralized or distributed to make data to effectively reach the destination. In wired networks, routers and switches define the network structure, but in wireless networks, and particularly in WSNs where hundreds or thousands of nodes have to be organized without any specific device to perform organization, the nodes themselves have to implement efficient self-organization

Self-organization in WSNs covers several tasks such as topology discovering, medium access control, data routing, and specific application controls. Self-organization can be defined as *the execution of local tasks by the individuals that take part in the network in order to get a global objective*

One of the most important tasks in self-organization in WSNs is routing, since it allows the network to stablish the routes necessary to correct and efficiently deliver network data to the

The special features of WSNs make that the development of routing schemes for this kind of networks must consider the following aspects (Pantazis et al., 2009; Yang & Mohammed,

*without using any centralized control* (Zvikhachevskaya & Mihaylova, 2009).

destination in a reliable manner (Royer & Toh, 1999).

levels, while ZigBee defines network layer and application framework (see Fig. 3).

**2.3.2 ZigBee**

below.

mechanisms.

2010):

Fig. 2. WSN node attached to a vine in the Wisevine project.

communications. Self-organization is an important issue in the world of sensor networks that ensures the correct operation of the networks and its efficiency.

#### **2.3 Architectures**

Architectures in WSNs are defined with the objective of organize protocols and communication services that can be executed by sensor nodes. This structure helps developers to create products that are completely functional when combining with other protocols, services and devices in the system (Forouzan, 2006).

Wireless sensor networks have adopted (with some changes) the five-layer architecture used in TCP/IP networks, as a result of the simplification of the OSI architecture. The most important changes are related to the inter-layer communication. While in TCP/IP there exists several interfaces that allow inter-layer communications, the architectures for WSNs incorporate global services to allow transparent inter-layer communication. The most popular architectures used in the field of sensor networks are 6LoWPAN and ZigBee.

#### **2.3.1 6LoWPAN**

*IPv6 over Low power Wireless Personal Area Networks* (Z. Shelby and C. Bormann, 2009) is an architecture that defines the use of IPv6 addressing for WSNs, allowing so the inclusion in the global network, favouring the access to network nodes from everywhere. Same as ZigBee, 6LoWPAN uses IEEE 802.15.4 for the definition of physical and medium access layers, while in the network layer it uses IPv6 addressing adapted to WSNs by using the LowPAN layer, that provides encapsulation and the necessary methods to allow the co-existence of 802.15.4 and IPv6. Transport layer can use UDP or ICMP, depending on the requirements of the particular application.

Fig. 3. ZigBee protocol stack.

#### **2.3.2 ZigBee**

4 Will-be-set-by-IN-TECH

communications. Self-organization is an important issue in the world of sensor networks

Architectures in WSNs are defined with the objective of organize protocols and communication services that can be executed by sensor nodes. This structure helps developers to create products that are completely functional when combining with other protocols,

Wireless sensor networks have adopted (with some changes) the five-layer architecture used in TCP/IP networks, as a result of the simplification of the OSI architecture. The most important changes are related to the inter-layer communication. While in TCP/IP there exists several interfaces that allow inter-layer communications, the architectures for WSNs incorporate global services to allow transparent inter-layer communication. The most popular

*IPv6 over Low power Wireless Personal Area Networks* (Z. Shelby and C. Bormann, 2009) is an architecture that defines the use of IPv6 addressing for WSNs, allowing so the inclusion in the global network, favouring the access to network nodes from everywhere. Same as ZigBee, 6LoWPAN uses IEEE 802.15.4 for the definition of physical and medium access layers, while in the network layer it uses IPv6 addressing adapted to WSNs by using the LowPAN layer, that provides encapsulation and the necessary methods to allow the co-existence of 802.15.4 and IPv6. Transport layer can use UDP or ICMP, depending on the requirements of the particular

architectures used in the field of sensor networks are 6LoWPAN and ZigBee.

Fig. 2. WSN node attached to a vine in the Wisevine project.

services and devices in the system (Forouzan, 2006).

**2.3 Architectures**

**2.3.1 6LoWPAN**

application.

that ensures the correct operation of the networks and its efficiency.

The ZigBee Alliance *ZigBee Specification, ZigBee Alliance* (2011) and the IEEE 802.15.4 *IEEE Standard for Part 15.4: Wireless Medium Access Control (MAC) and Physical Layer (PHY) specifications for Low-Rate Wireless Personal Area Networks (WPANS)* (2011) Task Group are leading the efforts to define a standard protocol stack for the implementation of wireless sensor networks. IEEE 802.15.4 is focused on the standardization of the MAC and physical levels, while ZigBee defines network layer and application framework (see Fig. 3).

The ZigBee network layer includes three different topologies, namely, tree, mesh, and cluster-based topologies. This chapter is focused on mesh topology networks for which ZigBee uses the Ad hoc On demand Distance Vector protocol (AODV), that will be detailed below.

#### **2.4 Self-organization and routing**

The correct operation of both wired and wireless networks requires some kind of network organization. Most of networking systems follow some kind of organization, well centralized or distributed to make data to effectively reach the destination. In wired networks, routers and switches define the network structure, but in wireless networks, and particularly in WSNs where hundreds or thousands of nodes have to be organized without any specific device to perform organization, the nodes themselves have to implement efficient self-organization mechanisms.

Self-organization in WSNs covers several tasks such as topology discovering, medium access control, data routing, and specific application controls. Self-organization can be defined as *the execution of local tasks by the individuals that take part in the network in order to get a global objective without using any centralized control* (Zvikhachevskaya & Mihaylova, 2009).

One of the most important tasks in self-organization in WSNs is routing, since it allows the network to stablish the routes necessary to correct and efficiently deliver network data to the destination in a reliable manner (Royer & Toh, 1999).

The special features of WSNs make that the development of routing schemes for this kind of networks must consider the following aspects (Pantazis et al., 2009; Yang & Mohammed, 2010):

Fig. 4. AODV route discovery example.

and the destination node (D).

**3.1 AODV drawbacks**

costs to complete routing tasks.

destination. In this case, it will send (unicast) an RREP to the originator of the received RREQ. Otherwise, the intermediate node will save the request in order to forward an eventual RREP, and the RREQ will be re-broadcast if TTL (Time-To-Live) value is greater than zero. Figure 4 shows an example of the messages sent during route discovery between the source node (A)

Fuzzy Logic Applied to Decision Making in Wireless Sensor Networks 227

To control network-wide broadcasts of RREQs, the source node uses an expanding ring search technique, which allows a search of increasingly larger areas of the network if a route to the destination is not found. In order to avoid loops and forwarding storms, both RREQ and RREP packets are forwarded just once unless an intermediate node receives an RREQ or RREP with the same source and destination addresses, but with a lower number of hops. In that case, it will be forwarded in order to discover the route with the lowest number of hops. Eventually, the source node will receive an RREP if there is a route to the destination. Then, buffered data

In the case of a link failure, implied nodes will generate an RRER message in order to notify

Due to its on-demand-based nature, AODV presents several problems that are mainly related to high packet drop ratios and high routing overheads (Alshanyour & Baroudi, 2008). These problems cause packet loss, collisions, high end-to-end delay and high latency, among others. • **Packet overhead**: AODV requires an enormous number of packets to complete path discovery and perform routing tasks (Lin, 2005; Sklyarenko, 2006). RREQ broadcasts represent a high network load, and this load is increased when packets have to be re-injected due to high channel occupancy and collisions. As the node density increases, the number of messages sent and received per node appears to increase quadratically (Sklyarenko, 2006). This occurs because when nodes broadcast RREQ messages, those messages are received by more nodes, and these nodes occupy the channel rebroadcasting them. As more nodes come together, the channel scheduling becomes more difficult. • **Redundant discovery**: routes frequently become saturated causing blocks, thereby leading to new route discoveries. These route discoveries increase the routing overhead, thus aggravating the problem (Pirzada & et al., 2007). Moreover, the path discovery overhead and the routing overhead are sometimes very high, with the consequent time and energy

packets can be sent to the destination node using the newly-discovered route.

communicating nodes about the invalidation of the routes using that link.


The consideration of these factors will ensure the achievement of the routing protocols, but it is important to consider some requirements such as scalability, fault tolerance, efficiency or quality of service, in order to get the desired result when using the routing approach.

Next, AODV routing protocol is analysed in order to illustrate its main features and drawbacks.

#### **3. Ad-hoc On demand Distance Vector routing (AODV)**

AODV is a pure on-demand routing protocol which bases route discovery on a route request and route reply query cycle and the metric used is the number of hops from the source to the destination. In general terms, when a source node aims to send data to a destination node, the source broadcasts a route-request packet in order to discover a route to the destination. Intermediate nodes will forward the route-request, and eventually, any node which has a route to the destination or the destination itself will reply (unicast) with a route-reply message to the source. Once the source has received the route-reply, it is ready to send data to the destination. Routes are maintained and if any error occurs during the route valid time (or lifetime), a route-error message is propagated in order to avoid the use of broken links and out-of-date routes.

Messages used in AODV during route discovery and maintenance processes are:


In AODV, the route discovery process starts when a source node intends to communicate with a destination node. If the route is unknown, data packets are buffered, and the source node broadcasts an RREQ intended for the destination node. A node receiving an RREQ will verify the destination address to check if it is the destination node, or if it has a route to the

Fig. 4. AODV route discovery example.

6 Will-be-set-by-IN-TECH

• **Resource limitations:** restrictions such as available energy, memory and processing capabilities should be considered in order to extend, as much as possible, the network

• **Node heterogeneity:** it is possible the coexistence of different node models in the same network. So, the routing protocol should solve the problems that can arise when nodes

• **Transmission medium:** problems regarding the wireless channel such as interferences,

• **Coverage and connectivity:** since the node coverage is limited, the connectivity of all the network must be ensured, avoiding node isolation, and enabling multi-hop

The consideration of these factors will ensure the achievement of the routing protocols, but it is important to consider some requirements such as scalability, fault tolerance, efficiency or

Next, AODV routing protocol is analysed in order to illustrate its main features and

AODV is a pure on-demand routing protocol which bases route discovery on a route request and route reply query cycle and the metric used is the number of hops from the source to the destination. In general terms, when a source node aims to send data to a destination node, the source broadcasts a route-request packet in order to discover a route to the destination. Intermediate nodes will forward the route-request, and eventually, any node which has a route to the destination or the destination itself will reply (unicast) with a route-reply message to the source. Once the source has received the route-reply, it is ready to send data to the destination. Routes are maintained and if any error occurs during the route valid time (or lifetime), a route-error message is propagated in order to avoid the use of broken links and

• **Route Request (RREQ)**: this kind of messages are used to discover network routes. An RREQ contains: ID, source and destination addresses, sequence number, hop count, time-to-live (TTL), and control flags. RREQ ID, combined with the source address,

• **Route Reply (RREP)**: it is used to answer route-request messages. It contains source and destination addresses, route lifetime, sequence number, hop count and control flags. • **Route Error (RRER)**: these messages are used to notify of link failures, and avoid their use. They contain the addresses and corresponding destination sequence number of all active destinations that have become unreachable due to the link failure. A node receiving an

In AODV, the route discovery process starts when a source node intends to communicate with a destination node. If the route is unknown, data packets are buffered, and the source node broadcasts an RREQ intended for the destination node. A node receiving an RREQ will verify the destination address to check if it is the destination node, or if it has a route to the

quality of service, in order to get the desired result when using the routing approach.

Messages used in AODV during route discovery and maintenance processes are:

RRER message, will invalidate the corresponding entries in its routing table.

lifetime without overloading the network and the nodes themselves.

with different hardware or radio interface have to collaborate.

signal attenuation or collisions must be considered.

**3. Ad-hoc On demand Distance Vector routing (AODV)**

communication if necessary.

drawbacks.

out-of-date routes.

uniquely identifies an RREQ.

destination. In this case, it will send (unicast) an RREP to the originator of the received RREQ. Otherwise, the intermediate node will save the request in order to forward an eventual RREP, and the RREQ will be re-broadcast if TTL (Time-To-Live) value is greater than zero. Figure 4 shows an example of the messages sent during route discovery between the source node (A) and the destination node (D).

To control network-wide broadcasts of RREQs, the source node uses an expanding ring search technique, which allows a search of increasingly larger areas of the network if a route to the destination is not found. In order to avoid loops and forwarding storms, both RREQ and RREP packets are forwarded just once unless an intermediate node receives an RREQ or RREP with the same source and destination addresses, but with a lower number of hops. In that case, it will be forwarded in order to discover the route with the lowest number of hops. Eventually, the source node will receive an RREP if there is a route to the destination. Then, buffered data packets can be sent to the destination node using the newly-discovered route.

In the case of a link failure, implied nodes will generate an RRER message in order to notify communicating nodes about the invalidation of the routes using that link.

#### **3.1 AODV drawbacks**

Due to its on-demand-based nature, AODV presents several problems that are mainly related to high packet drop ratios and high routing overheads (Alshanyour & Baroudi, 2008). These problems cause packet loss, collisions, high end-to-end delay and high latency, among others.


that, they use as input variables the temperature, humidity, fan speed, and engine speed. The experiments show good results compared to a traditional control system based on discrete

Fuzzy Logic Applied to Decision Making in Wireless Sensor Networks 229

An example of the use of fuzzy logic in routing for WSNs is LEACH-FL (Ran et al., 2010), where the selection of cluster-heads is based on several variables: node battery level, node density and distance to the base station. The experiments show that the use of fuzzy logic helps to reduce the energy consumption, so extending the overall network lifetime. Another example of fuzzy logic in WSN routing is (Ortiz et al., 2011) where the metric of the Tree Routing protocol used in ZigBee is replaced with the output of a fuzzy-logic based mechanism that allows a reduction in the path length, in the network discovery time and in the number

In summary, the fuzzy logic is a powerful tool to be used in WSN approaches, since it provides effective parameter combination, and it is able to be executed in the low-resourced nodes that compose these networks. The next section details AODV-FL, a routing approach for wireless sensor networks that makes use of the fuzzy logic to evaluate several parameters that are

The use of fuzzy logic in the decision-making processes is detailed herein in order to select the best nodes to be part of the routes, and the incorporation of a timer when a new RREQ is received, to be able, if necessary, to evaluate several RREQs received (with the same ID and sequence number) and just forward the best of all those, instead of sometimes forwarding a worse RREQ and later a better one, as the traditional AODV does. With this timer we aim to reduce the number of messages used to discover routes, and so the network congestion

The lack of an efficient metric to evaluate node conditions in AODV has been solved by the definition of a new metric based on the combination of different node and network parameters by using a fuzzy-logic system. The idea is to specify the input parameters in natural language and, with the help of a fuzzy-rule set, to define the relationship among different inputs with the output, which represents the suitability or quality of a node to be selected as a part of the

• **Number of hops**: this is the length of the path. In general, a lower number of hops will represent a better route, but this is not true at all, since it is possible that some nodes in the route have low battery or bad Received Signal Strength Indicator (RSSI), so it is very important to consider more variables to decide the route. This input fuzzy set is shown in Fig. 5a. The maximum number of hops observed in our experiments has been 5. Fuzzy sets have been declared to deal with any extreme situation that can occur during the execution.

• **Local Battery level**: this parameter must be considered in order to avoid nodes with low battery taking part in data paths since they can cause failures in communication. Route construction considering nodes with high energy levels will help to save the energy of low-battery nodes and will cooperate to balance network lifetime. Moreover, the consideration of the battery level will ensure data transmission, preventing nodes in the

These fuzzy sets can be customized depending on each particular network size.

**5. Ad-hoc On demand Distance Vector Routing with Fuzzy Logic (AODV-FL)**

temperature values.

of forwarding nodes.

incoming route.

considered during the route-creation process.

caused by this high number of messages.

The input parameters to be considered are:


In order to solve some of these problems, next section details the use of fuzzy logic in WSNs, as a backgraund of the proposal detailed in Section 5

#### **4. Fuzzy Logic and Wireless Sensor Networks**

In the literature, there exists several techniques oriented to improve the performance of routing approaches for WSNs. Most of these techniques are focused on changing the metric used to optimize parameters in order to determine the best path between source and destination, reduce the number of packets used, or reduce the end-to-end delay, among others.

The use of fuzzy logic to optimize the metric used in routing approaches for WSNs is a promising technique since it allows combine and evaluate diverse parameters in an efficient manner. Moreover, several proposals have shown that the use of fuzzy logic in this kind of networks is a good choice due to the execution requirements can be easily supported by sensor nodes, while it is able to improve the overall network performance.

Fuzzy logic is used in (Bacour et al., 2010) to perform link quality estimation. The system takes as input the information about link capacity to transport information, asymmetry, stability and channel quality. The experiments in a network in which all nodes are reachable from the base station show improvements in terms of reliability and stability.

In (Wang et al., 2009) is presented a method based on fuzzy logic and implemented in ZigBee nodes, with the aim of reducing the on/off frequency of an air conditioner system. To do 8 Will-be-set-by-IN-TECH

• **High route discovery delay**: as a reactive protocol, AODV has an evident weakness: its latency, since routes are discovered on demand. The route discovery process can take some time and this delay can be increased due to problems in the medium access, such as busy channel and collisions. The time taken by the network to create routes exhibits cubic growth in relation to the number of network nodes (Sklyarenko, 2006). AODV's end-to-end delay is also a weakness of this protocol since it becomes very high when a big proportion of the network nodes have to send messages. This problem is caused by collisions during

the routing discovery process, and during data forwarding (Nefzy & Song, 2007). • **High memory demand**: along with time, memory is also critical and AODV requires all nodes to reserve sufficiently large memory spaces to store possible routing entries for active sources and destinations (Lin, 2005; Ramachandran et al., 2005). This is a problem that limits scalability in WSNs and is due to nodes being resource constrained (Manjula et al., 2008). The throughput of AODV is compromised due to high packet loss (Pirzada & et al., 2007). Since data delivery is a critical issue for some applications such as health and

• **Duplicated messages**: the route discovery process also has some problems due to the absence of a delay between receiving and forwarding discovery packets. For example, a node that has just forwarded an RREQ from a source node, may receive the same RREQ with a lower number of hops, and it will have to forward it again, thus increasing energy

• **Deficient metric**: another problem in AODV, is the metric used to make routing decisions. AODV forms routes using only the number of hops as a metric. Even though one may agree that AODV can always choose the route that minimizes the delay (Boughanmi & Song, 2007), it does not take into account other important parameters, such as available node energy, route traffic, or the signal strength of the received packets, among others.

In order to solve some of these problems, next section details the use of fuzzy logic in WSNs,

In the literature, there exists several techniques oriented to improve the performance of routing approaches for WSNs. Most of these techniques are focused on changing the metric used to optimize parameters in order to determine the best path between source and destination, reduce the number of packets used, or reduce the end-to-end delay, among others. The use of fuzzy logic to optimize the metric used in routing approaches for WSNs is a promising technique since it allows combine and evaluate diverse parameters in an efficient manner. Moreover, several proposals have shown that the use of fuzzy logic in this kind of networks is a good choice due to the execution requirements can be easily supported by sensor

Fuzzy logic is used in (Bacour et al., 2010) to perform link quality estimation. The system takes as input the information about link capacity to transport information, asymmetry, stability and channel quality. The experiments in a network in which all nodes are reachable from the base

In (Wang et al., 2009) is presented a method based on fuzzy logic and implemented in ZigBee nodes, with the aim of reducing the on/off frequency of an air conditioner system. To do

monitoring, packet loss has to be minimized.

as a backgraund of the proposal detailed in Section 5

**4. Fuzzy Logic and Wireless Sensor Networks**

nodes, while it is able to improve the overall network performance.

station show improvements in terms of reliability and stability.

consumption and network traffic.

that, they use as input variables the temperature, humidity, fan speed, and engine speed. The experiments show good results compared to a traditional control system based on discrete temperature values.

An example of the use of fuzzy logic in routing for WSNs is LEACH-FL (Ran et al., 2010), where the selection of cluster-heads is based on several variables: node battery level, node density and distance to the base station. The experiments show that the use of fuzzy logic helps to reduce the energy consumption, so extending the overall network lifetime. Another example of fuzzy logic in WSN routing is (Ortiz et al., 2011) where the metric of the Tree Routing protocol used in ZigBee is replaced with the output of a fuzzy-logic based mechanism that allows a reduction in the path length, in the network discovery time and in the number of forwarding nodes.

In summary, the fuzzy logic is a powerful tool to be used in WSN approaches, since it provides effective parameter combination, and it is able to be executed in the low-resourced nodes that compose these networks. The next section details AODV-FL, a routing approach for wireless sensor networks that makes use of the fuzzy logic to evaluate several parameters that are considered during the route-creation process.

#### **5. Ad-hoc On demand Distance Vector Routing with Fuzzy Logic (AODV-FL)**

The use of fuzzy logic in the decision-making processes is detailed herein in order to select the best nodes to be part of the routes, and the incorporation of a timer when a new RREQ is received, to be able, if necessary, to evaluate several RREQs received (with the same ID and sequence number) and just forward the best of all those, instead of sometimes forwarding a worse RREQ and later a better one, as the traditional AODV does. With this timer we aim to reduce the number of messages used to discover routes, and so the network congestion caused by this high number of messages.

The lack of an efficient metric to evaluate node conditions in AODV has been solved by the definition of a new metric based on the combination of different node and network parameters by using a fuzzy-logic system. The idea is to specify the input parameters in natural language and, with the help of a fuzzy-rule set, to define the relationship among different inputs with the output, which represents the suitability or quality of a node to be selected as a part of the incoming route.

The input parameters to be considered are:


**Nhops Bat**. **RSSI Output Nhops Bat**. **RSSI Output Nhops Bat. RSSI Output** Low Low Low **Low** Med Low Low **Low** High Low Low **Low** Low Low Med **Low** Med Low Med **Low** High Low Med **Low** Low Low High **Med** Med Low High **Med** High Low High **Med** Low Med Low **Low** Med Med Low **Low** High Med Low **Low** Low Med Med **Med** Med Med Med **Med** High Med Med **Low** Low Med High **High** Med Med High **Med** High Med High **Med** Low High Low **Med** Med High Low **Low** High High Low **Low** Low High Med **High** Med High Med **Med** High High Med **Med** Low High High **High** Med High High **High** High High High **Med**

Fuzzy Logic Applied to Decision Making in Wireless Sensor Networks 231

forward its request/reply packet, with the objective of reducing packet overhead and energy

The input parameters, sets and rules shown herein, are just an example for the particular application and network model used in our experiments. Note that both fuzzy sets and rules, as well as considered parameters, can be customized depending on the application

In AODV-FL, a node receiving an RREQ calculates the fuzzy-logic value associated to that RREQ, and if it is the first RREQ received (no RREQ with the same ID and sequence number has been received), it starts a timer. During the duration of the timer, if the node receives more RREQs with the same ID and sequence number, the stored request will be updated if the calculated FL-value for the received RREQ is higher than the one stored. When the timer

The destination node, or any intermediate node having a route to the destination, will reply

Flow charts for AODV and AODV-FL are shown in Figs. 7 and 8. There are two main differences between both proposals: first, the change of metric, the number of hops used in AODV, for the output of the FL-evaluation process in AODV-FL; and second, the use of a timer to allow the reception (if necessary) of several RREQs from the same source node, and select the best (fuzzy-logic evaluation based) RREQ to be forwarded, thus avoiding multiple forwarding for the same RREQ. This event is frequent in AODV when using a realistic MAC protocol, because sometimes a node may receive first an RREQ with *numhops* = *x* and later another RREQ with *numhops* = *x* − *n*, and both will be forwarded. In contrast, the timer implemented in AODV-FL allows nodes to wait for more RREQs (with the same ID and sequence number) when the first one is received. This timer is randomly calculated by considering one-hop packet delivery time and the *MaxBackOff* parameter from the MAC layer.

expires, the node will forward the received RREQ with the highest FL value.

with an RREP to the best RREQ received (for a given ID and sequence number).

• Reduce the number of packets sent, so reducing the global energy consumption.

• Maintain routing table size, not making the use of extra memory space.

• Improve route formation by selecting, at each hop, the best available node, ensuring route

requirements, node features, network size and capabilities.

Table 1. Fuzzy rule base

With these premises we aim to:

stability and avoiding data loss.

consumption.

(a) Fuzzy Sets for RSSI. (b) Output Fuzzy Sets.

route from running out of battery. Fuzzy sets for battery level are shown in Fig. 5b. The X-axis represents (as %) the remaining battery of the node.

• **RSSI (Received Signal Strength Indicator)**: the strength of the received signal is an indicator of the quality of communications between two nodes. In order to ensure quality communications and prevent data loss, data paths will consist of nodes that are able to communicate with a certain level of signal quality. Figure 6a shows the fuzzy sets declared for this variable. The X-axis represents (as %) the strength of the received signal.

The output of the fuzzy system (see Fig. 6b) represents the suitability of a node to be considered for inclusion in the route.

The geometric pattern of triangles is commonly used to determine the appropriate membership functions and control rules in many theory applications (Wang et al., 2009). In this paper, the geometric pattern of triangles to define input and output variables has been adopted.

Input and output sets are combined through a set of rules in order to obtain the corresponding output. Table 1 depicts the fuzzy-rule base used in the experiments. The objective of the fuzzy rules is to serve as a basis to determine, during the route discovery process, the best node to


Table 1. Fuzzy rule base

10 Will-be-set-by-IN-TECH

Membership Degree

(a) Fuzzy Sets for Number of Hops. (b) Fuzzy Sets for Battery Level.

Membership Degree

(a) Fuzzy Sets for RSSI. (b) Output Fuzzy Sets.

route from running out of battery. Fuzzy sets for battery level are shown in Fig. 5b. The

• **RSSI (Received Signal Strength Indicator)**: the strength of the received signal is an indicator of the quality of communications between two nodes. In order to ensure quality communications and prevent data loss, data paths will consist of nodes that are able to communicate with a certain level of signal quality. Figure 6a shows the fuzzy sets declared

The output of the fuzzy system (see Fig. 6b) represents the suitability of a node to be

The geometric pattern of triangles is commonly used to determine the appropriate membership functions and control rules in many theory applications (Wang et al., 2009). In this paper, the geometric pattern of triangles to define input and output variables has been

Input and output sets are combined through a set of rules in order to obtain the corresponding output. Table 1 depicts the fuzzy-rule base used in the experiments. The objective of the fuzzy rules is to serve as a basis to determine, during the route discovery process, the best node to

for this variable. The X-axis represents (as %) the strength of the received signal.

% of Remaining Battery

Node Goodness

Number of hops from source/to

Received Signal Strength Indicator (%)

X-axis represents (as %) the remaining battery of the node.

Fig. 6. Input and Output Fuzzy sets.

considered for inclusion in the route.

2345

Membership Degree

0

Membership Degree

adopted.

Fig. 5. Input Fuzzy sets.

forward its request/reply packet, with the objective of reducing packet overhead and energy consumption.

The input parameters, sets and rules shown herein, are just an example for the particular application and network model used in our experiments. Note that both fuzzy sets and rules, as well as considered parameters, can be customized depending on the application requirements, node features, network size and capabilities.

In AODV-FL, a node receiving an RREQ calculates the fuzzy-logic value associated to that RREQ, and if it is the first RREQ received (no RREQ with the same ID and sequence number has been received), it starts a timer. During the duration of the timer, if the node receives more RREQs with the same ID and sequence number, the stored request will be updated if the calculated FL-value for the received RREQ is higher than the one stored. When the timer expires, the node will forward the received RREQ with the highest FL value.

The destination node, or any intermediate node having a route to the destination, will reply with an RREP to the best RREQ received (for a given ID and sequence number).

Flow charts for AODV and AODV-FL are shown in Figs. 7 and 8. There are two main differences between both proposals: first, the change of metric, the number of hops used in AODV, for the output of the FL-evaluation process in AODV-FL; and second, the use of a timer to allow the reception (if necessary) of several RREQs from the same source node, and select the best (fuzzy-logic evaluation based) RREQ to be forwarded, thus avoiding multiple forwarding for the same RREQ. This event is frequent in AODV when using a realistic MAC protocol, because sometimes a node may receive first an RREQ with *numhops* = *x* and later another RREQ with *numhops* = *x* − *n*, and both will be forwarded. In contrast, the timer implemented in AODV-FL allows nodes to wait for more RREQs (with the same ID and sequence number) when the first one is received. This timer is randomly calculated by considering one-hop packet delivery time and the *MaxBackOff* parameter from the MAC layer.

With these premises we aim to:


RREQ received ID = X, SN = Y

Start �mer if first RREQ received with ID = = X and SN == Y

> Is already in table?

Yes

Fuzzy Logic Applied to Decision Making in Wireless Sensor Networks 233

Yes No

Create new table entry

Timer expired (ID=X and SN=Y)

No

Fig. 8. AODV-FL decision flowchart.

Am I the des�na�on or do I have a route?

Yes

No

So nodes 3 and 4 compete to forward the RREQ. Let's suppose once again that node 4 owns the channel and forwards the RREQ which is received by nodes 2, 3 and *DEST*. Nodes 2 and 3 discard it and *DEST* generates an RREP and sends it to node 4. This RREP will be forwarded by nodes 2 and 1 until it reaches *SOURCE*. Now, node 3 finds the channel free, so it forwards the RREQ that received from *SOURCE*. Nodes 1, 2 and 4 receive this packet. Nodes 1 and 2 discard it since it does not improve their hop counts, and node 4 forwards it since it improves the hop count (previously 3, now 2). *DEST* receives this RREQ and generates a new RREP, because it improves the stored hop count. The new route now has 3 hops instead of the 4 hops of the previous route. Then (not shown) Node 4 will forward

• **AODV-FL**: (shown in Fig. 9c) nodes 1 and 3 receive the RREQ from *SOURCE* and both start a timer in order to wait to receive more RREQs with equal ID and sequence number. Let's suppose that the timer in node 1 finishes first(note that timers are set with a random time proportional to the number of different RREQs received). So node 1 forwards the packet. Nodes 2 and 3 receive the RREQ; node 3 discards it since it does not improve its

Send RREP

the RREP to 3, which will forward it to *SOURCE* (not shown in Fig. 9b).

Yes

Drop RREQ ID = X, SN = Y

Improves FL stored value?

No

Update table

Forward RREQ (ID = X; SN = Y)

Fig. 7. AODV decision flowchart.

• Provide adaptability: AODV-FL is able to deal with different networks in various applications, it is just necessary to tune the fuzzy parameters to be used, as well as the fuzzy sets and rules.

Figure 9 shows an example of message exchange during a part of route discovery for both AODV and AODV-FL. The topology used in this example is shown in Fig. 9a, in which the dotted line shows the connections in terms of the coverage of each node. *SOURCE* node aims to send data to *DEST* node, and broadcasts an RREQ. Lett's detail the operation of AODV, and our proposal, AODV-FL:

• **AODV**: (shown in Fig. 9b) nodes 1 and 3 receive the RREQ from *SOURCE* and both aim to forward it. Let's suppose that CSMA/CA (implemented in MAC layer) makes node 1 own the channel, so it forwards the RREQ, and node 3 buffers it to forward it later. Nodes 2 and 3 receive that packet, and just node 2 will forward it since node 3 has buffered an RREQ with a lower number of hops. Suppose that node 2 finds the channel free, and forwards the RREQ. Nodes 1, 3 and 4 receive it. Nodes 1 and 3 discards the packet since it does not improve the hop count stored for that RREQ. Remember that node 3 has an RREQ buffered. 12 Will-be-set-by-IN-TECH

Yes

Yes No

Drop RREQ ID = X, SN = Y

Improves hop count stored value?

No

Update table

Forward RREQ (ID = X; SN = Y)

RREQ received ID = X, SN = Y

Is already in table?

Create new table entry

Am I the des�na�on or do I have a route?

No

• Provide adaptability: AODV-FL is able to deal with different networks in various applications, it is just necessary to tune the fuzzy parameters to be used, as well as the

Figure 9 shows an example of message exchange during a part of route discovery for both AODV and AODV-FL. The topology used in this example is shown in Fig. 9a, in which the dotted line shows the connections in terms of the coverage of each node. *SOURCE* node aims to send data to *DEST* node, and broadcasts an RREQ. Lett's detail the operation of AODV,

• **AODV**: (shown in Fig. 9b) nodes 1 and 3 receive the RREQ from *SOURCE* and both aim to forward it. Let's suppose that CSMA/CA (implemented in MAC layer) makes node 1 own the channel, so it forwards the RREQ, and node 3 buffers it to forward it later. Nodes 2 and 3 receive that packet, and just node 2 will forward it since node 3 has buffered an RREQ with a lower number of hops. Suppose that node 2 finds the channel free, and forwards the RREQ. Nodes 1, 3 and 4 receive it. Nodes 1 and 3 discards the packet since it does not improve the hop count stored for that RREQ. Remember that node 3 has an RREQ buffered.

Send RREP

Fig. 7. AODV decision flowchart.

fuzzy sets and rules.

and our proposal, AODV-FL:

Yes

Fig. 8. AODV-FL decision flowchart.

So nodes 3 and 4 compete to forward the RREQ. Let's suppose once again that node 4 owns the channel and forwards the RREQ which is received by nodes 2, 3 and *DEST*. Nodes 2 and 3 discard it and *DEST* generates an RREP and sends it to node 4. This RREP will be forwarded by nodes 2 and 1 until it reaches *SOURCE*. Now, node 3 finds the channel free, so it forwards the RREQ that received from *SOURCE*. Nodes 1, 2 and 4 receive this packet. Nodes 1 and 2 discard it since it does not improve their hop counts, and node 4 forwards it since it improves the hop count (previously 3, now 2). *DEST* receives this RREQ and generates a new RREP, because it improves the stored hop count. The new route now has 3 hops instead of the 4 hops of the previous route. Then (not shown) Node 4 will forward the RREP to 3, which will forward it to *SOURCE* (not shown in Fig. 9b).

• **AODV-FL**: (shown in Fig. 9c) nodes 1 and 3 receive the RREQ from *SOURCE* and both start a timer in order to wait to receive more RREQs with equal ID and sequence number. Let's suppose that the timer in node 1 finishes first(note that timers are set with a random time proportional to the number of different RREQs received). So node 1 forwards the packet. Nodes 2 and 3 receive the RREQ; node 3 discards it since it does not improve its

*Parameter Value* max MAC Frame Size 80 bytes MAC Frame Overhead 14 bytes MAC Buffer Size 32 frames

Fuzzy Logic Applied to Decision Making in Wireless Sensor Networks 235

min Exponential Backoff 3 max Exponential Backoff 5 max CSMA Backoffs 4 max Frame Retries 3 Table 2. MAC parameters used in the experiments with AODV, AODV-FL and AODV-ETX

In order to evaluate the performance of our proposal, we have implemented AODV, AODV-FL, AODV-ETX (AODV using ETX-based metric), and CSMA/CA in the Omnet++ (*Omnet++ Network Simulation Framework*, 2011) module for wireless sensor simulation. The use of a realistic MAC protocol will provide us with reliable results in order to include our

In AODV-ETX Ni et al. (2008), the hop-count metric is replaced with a new metric based on expected transmissions, ETX (Expected Transmissions Count) Couto et al. (2003) aims to find high-throughput paths on multihop wireless networks, by minimizing the expected total number of packet transmissions required to successfully deliver a packet to the ultimate

In the experiments, nodes decide whether to discover a route and send data to a random destination with a probability of 25%. Routes are established on demand and the experiments consists on the sending nodes executing the discovery process and sending one data packet. Nodes are deployed randomly with a separation between nodes which varies between 1 and 50 meters. The number of nodes varies from 25 to 200, and each experiment has been executed

In order to ensure route discovery, and taking into account that CSMA/CA is used to perform channel access, when the MAC layer reports *MAX NUMBER OF BACKOFF* or *MAX FRAME RETRIES* achieved for a particular packet, this packet will be re-injected by the network layer.

To make a fair comparison, the results for AODV-ETX do not show the process of ETX

The variables to be evaluated are: energy consumption, number of RREQ and RREP packets

The energy consumption is a key element in WSNs; energy saving is a key objective of protocols for this kind of networks. Figure 10 shows (as %) the average energy saving achieved by AODV-FL and AODV-ETX with respect to the original AODV. The energy consumption of AODV-FL and AODV-ETX have been normalized according to the energy

Table 2 shows the main MAC parameters used in the experiments.

sent, number of collisions, end-to-end delay, and number of hops.

calculation which is carried out prior to the first RREQ send.

**6. Experiments**

destination.

**6.1 Results**

consumed in AODV.

50 times to get reliable results.

proposal in a real wireless sensor network.

Fig. 9. Message exchange example for AODV and AODV-FL.

FL-value, and node 2 starts a timer. Let's suppose that the timer of node 2 finishes before the one in node 3. So node 2 forwards the RREQ, which is received by nodes 1, 3, and 4. Node 1 discards it, since it has already forwarded that RREQ; node 3 discards it, and node 4 starts a timer. Now, the timer in node 3 finishes and it forwards the RREQ from *SOURCE*. Node 4 ignores it, due to as it does not improve the stored FL-value (node 2, 0.75). When the timer in node 4 expires, it forwards the RREQ. *DEST* receives the RREQ and generates an RREP for node 4. Node 4 will forward (not shown in Fig. 9b) the RREP to node 2 since the best RREQ received by node 4 came from node 2. Now the route has 4 hops instead of the 3 selected by AODV, but it is important to consider the low FL-value obtained by node 3, which may be a sign of packet loss.

The example shows the efficiency of route discovery with AODV-FL, which even selects routes with more hops but that are able to avoid data loss. AODV selected the shortest route, but node 3 may present battery or signal strength problems that cause packet loss, with the consequent energy consumption caused by re-injection. Besides the reliability of the routes created by AODV-FL, it is important to consider the energy saving achieved: only with six nodes, AODV-FL reduces the number of packets by 25%. This packet reduction will rise when the network size increases.


Table 2. MAC parameters used in the experiments with AODV, AODV-FL and AODV-ETX

#### **6. Experiments**

14 Will-be-set-by-IN-TECH

(a) Example topology

(b) AODV timeline. (c) AODV-FL timeline.

FL-value, and node 2 starts a timer. Let's suppose that the timer of node 2 finishes before the one in node 3. So node 2 forwards the RREQ, which is received by nodes 1, 3, and 4. Node 1 discards it, since it has already forwarded that RREQ; node 3 discards it, and node 4 starts a timer. Now, the timer in node 3 finishes and it forwards the RREQ from *SOURCE*. Node 4 ignores it, due to as it does not improve the stored FL-value (node 2, 0.75). When the timer in node 4 expires, it forwards the RREQ. *DEST* receives the RREQ and generates an RREP for node 4. Node 4 will forward (not shown in Fig. 9b) the RREP to node 2 since the best RREQ received by node 4 came from node 2. Now the route has 4 hops instead of the 3 selected by AODV, but it is important to consider the low FL-value

The example shows the efficiency of route discovery with AODV-FL, which even selects routes with more hops but that are able to avoid data loss. AODV selected the shortest route, but node 3 may present battery or signal strength problems that cause packet loss, with the consequent energy consumption caused by re-injection. Besides the reliability of the routes created by AODV-FL, it is important to consider the energy saving achieved: only with six nodes, AODV-FL reduces the number of packets by 25%. This packet reduction will rise when

Fig. 9. Message exchange example for AODV and AODV-FL.

obtained by node 3, which may be a sign of packet loss.

the network size increases.

In order to evaluate the performance of our proposal, we have implemented AODV, AODV-FL, AODV-ETX (AODV using ETX-based metric), and CSMA/CA in the Omnet++ (*Omnet++ Network Simulation Framework*, 2011) module for wireless sensor simulation. The use of a realistic MAC protocol will provide us with reliable results in order to include our proposal in a real wireless sensor network.

In AODV-ETX Ni et al. (2008), the hop-count metric is replaced with a new metric based on expected transmissions, ETX (Expected Transmissions Count) Couto et al. (2003) aims to find high-throughput paths on multihop wireless networks, by minimizing the expected total number of packet transmissions required to successfully deliver a packet to the ultimate destination.

In the experiments, nodes decide whether to discover a route and send data to a random destination with a probability of 25%. Routes are established on demand and the experiments consists on the sending nodes executing the discovery process and sending one data packet. Nodes are deployed randomly with a separation between nodes which varies between 1 and 50 meters. The number of nodes varies from 25 to 200, and each experiment has been executed 50 times to get reliable results.

In order to ensure route discovery, and taking into account that CSMA/CA is used to perform channel access, when the MAC layer reports *MAX NUMBER OF BACKOFF* or *MAX FRAME RETRIES* achieved for a particular packet, this packet will be re-injected by the network layer. Table 2 shows the main MAC parameters used in the experiments.

To make a fair comparison, the results for AODV-ETX do not show the process of ETX calculation which is carried out prior to the first RREQ send.

#### **6.1 Results**

The variables to be evaluated are: energy consumption, number of RREQ and RREP packets sent, number of collisions, end-to-end delay, and number of hops.

The energy consumption is a key element in WSNs; energy saving is a key objective of protocols for this kind of networks. Figure 10 shows (as %) the average energy saving achieved by AODV-FL and AODV-ETX with respect to the original AODV. The energy consumption of AODV-FL and AODV-ETX have been normalized according to the energy consumed in AODV.

 0 0.5 1 1.5 2 2.5 3 3.5 4

last data packet arrives to the destination.

Time (s)

AODV AODV−FL AODV−ETX

AODV AODV−FL AODV−ETX

Collisions

Fig. 12. Number of collisions.

Fig. 13. End-to-end delay.

60% more than AODV-FL.

0 50 100 150 200

Fuzzy Logic Applied to Decision Making in Wireless Sensor Networks 237

0 50 100 150 200

ones, require low end-to-end communication time, which includes route discovery, and data delivery. Figure 13 shows the average end-to-end delay since the first RREQ is sent until the

The delay introduced with the timer in AODV-FL is not a failing, because the high number of collisions makes AODV and AODV-ETX spend a lot of time re-injecting packets, around 40 to

Another important result is the number of hops. The example in Section 5 shows that AODV-FL may not select the route with lowest number of hops, while AODV does. In that example, AODV firstly selects a non-optimum route (in terms of the number of hops) and later the best route. Figure 14 shows the average number of hops (route length) for the routes created with the first RREP received by the source node for AODV, AODV-ETX and AODV-FL. The number of hops for the routes created when the source nodes receive the first RREP is higher for AODV with respect to AODV-FL. This is so because in AODV the source nodes may

Number of nodes

Number of nodes

Fig. 10. AODV-FL and AODV-ETX energy saving with respect to AODV energy consumption.

Fig. 11. Messages sent during route discovery phase for AODV, AODV-ETX and AODV-FL.

The energy consumed by AODV-FL is considerably lower than that consumed by AODV and AODV-ETX. This reduction will allow WSNs running AODV-FL to increase their lifetime. This energy saving is given due to the reduction in the number of packets sent during the route discovery phase. The number of RREQs and RREPs directly affects energy consumption, and is an important factor to be considered in the evaluation. Figure 11 depicts the average number of discovery messages sent by AODV (a), AODV-ETX (b) and AODV-FL (c) during the experiments.

The RREQ evaluation carried out by AODV-FL before packet forwarding, drastically reduces the number of discovery packets necessary to perform route creation. The high number of RREQs and RREPs sent in AODV and AODV-ETX, besides a higher energy consumption, it also leads to a high number of collisions. In AODV-FL, the RREQ evaluation, performed prior to forwarding, decreases the number of RREQ forwardings, and so reduces the number of collisions. The average number of collisions during the experiments is shown in Fig. 12, which confirms that the reduction in the number of RREQ and RREPs obtained by AODV-FL also reduces the number of collisions.

Collisions directly affect the communication delay since nodes have to re-inject collided packets. Networks with real-time requirements, such as industrial and building monitoring

Fig. 12. Number of collisions.

16 Will-be-set-by-IN-TECH

0 50 100 150 200

Number of nodes

25 50 75 100 125 150 175 200

RREQs RREPs

Number of Packets

25 50 75 100 125 150 175 200

Number of Nodes

Number of Nodes

(a) AODV. (b) AODV-ETX. (c) AODV-FL. Fig. 11. Messages sent during route discovery phase for AODV, AODV-ETX and AODV-FL.

The energy consumed by AODV-FL is considerably lower than that consumed by AODV and AODV-ETX. This reduction will allow WSNs running AODV-FL to increase their lifetime. This energy saving is given due to the reduction in the number of packets sent during the route discovery phase. The number of RREQs and RREPs directly affects energy consumption, and is an important factor to be considered in the evaluation. Figure 11 depicts the average number of discovery messages sent by AODV (a), AODV-ETX (b) and AODV-FL (c) during

The RREQ evaluation carried out by AODV-FL before packet forwarding, drastically reduces the number of discovery packets necessary to perform route creation. The high number of RREQs and RREPs sent in AODV and AODV-ETX, besides a higher energy consumption, it also leads to a high number of collisions. In AODV-FL, the RREQ evaluation, performed prior to forwarding, decreases the number of RREQ forwardings, and so reduces the number of collisions. The average number of collisions during the experiments is shown in Fig. 12, which confirms that the reduction in the number of RREQ and RREPs obtained by AODV-FL

Collisions directly affect the communication delay since nodes have to re-inject collided packets. Networks with real-time requirements, such as industrial and building monitoring

0

20

AODV AODV−FL AODV−ETX

Fig. 10. AODV-FL and AODV-ETX energy saving with respect to AODV energy

RREQs RREPs

Number of Packets

40

60

Energy consumption (%)

25 50 75 100 125 150 175 200

Number of Nodes

also reduces the number of collisions.

consumption.

RREQs RREPs

the experiments.

Number of Packets

80

100

Fig. 13. End-to-end delay.

ones, require low end-to-end communication time, which includes route discovery, and data delivery. Figure 13 shows the average end-to-end delay since the first RREQ is sent until the last data packet arrives to the destination.

The delay introduced with the timer in AODV-FL is not a failing, because the high number of collisions makes AODV and AODV-ETX spend a lot of time re-injecting packets, around 40 to 60% more than AODV-FL.

Another important result is the number of hops. The example in Section 5 shows that AODV-FL may not select the route with lowest number of hops, while AODV does. In that example, AODV firstly selects a non-optimum route (in terms of the number of hops) and later the best route. Figure 14 shows the average number of hops (route length) for the routes created with the first RREP received by the source node for AODV, AODV-ETX and AODV-FL.

The number of hops for the routes created when the source nodes receive the first RREP is higher for AODV with respect to AODV-FL. This is so because in AODV the source nodes may

or node density. The use of fuzzy logic in other layers, such as the MAC layer, will help to

Fuzzy Logic Applied to Decision Making in Wireless Sensor Networks 239

In summary, fuzzy logic is a powerful approach that has demonstrated to be effective when combining with other disciplines such as routing approaches for WSNs. The potential of fuzzy logic goes beyond traditional control systems and can be used on many research fields,

Alshanyour, A. M. & Baroudi, U. (2008). Bypass AODV: Improving Performance of Ad

Bacour, N., Koubaa, A., Youssef, H., Jamaa, M. B., do Rosario, D., Alves, M. & Becker,

Couto, D. D. J. D., Aguayo, D., Bicket, J. & Morris, R. (2003). A High-Throughtput Path Metric

*IEEE Standard for Part 15.4: Wireless Medium Access Control (MAC) and Physical Layer*

Manjula, S. H., Abhilash, C. N., Shaila, K., Venugopal, K. R. & Patniak, L. M. (2008).

Nefzy, B. & Song, Y. (2007). Performance Analysis and Improvement of ZigBee Routing

Ni, X., Lan, K. & Malaney, R. (2008). On the Performance of Expected Transmission Count

Ortiz, A. M., Olivares, T. & Orozco-Barbosa, L. (2011). Smart Routing Mechanism for Green

Pantazis, N. A., Nikolidakis, S. A. & V., D. D. (2009). Energy-efficient routing protocols

*Performance Evaluation Methodologies and Tools (VALUETOOLS)*. *Omnet++ Network Simulation Framework* (2011). http://www.omnetpp.org/.

Forouzan, B. A. (2006). *Transmisión de datos y redes de comunicaciones*, Mc. Grawn Hill.

http://standards.ieee.org/getieee802/download/802.15.4c-2009.pdf. Lin, C. (2005). AODV Routing Implementation for Scalable Wireless Ad-Hoc Network

Simulations (SWANS). JIST/SWANS, http://jist.ece.cornell.edu/.

*Proceedings of the European Conference on Wireless Sensor Networks (EWSN)*. Boughanmi, N. & Song, Y. (2007). Improvement of ZigBee Routing Protocol Including

Hoc On-Demand Distance Vector (AODV) Routing Protocol in Wireless Ad Hoc Networks, *Proceedings of the First International Conference on Ambient Media and Systems*

L. B. (2010). F-LQE: A Fuzzy Link Quality Estimator for Wireless Sensor Networks,

Energy and Delay Constraints, *Proceedings of the Junior Research Workshop on Real-Time*

for Multi-Hop Wireless Routing, *Proceedings of the 9th Annual International Conference*

*(PHY) specifications for Low-Rate Wireless Personal Area Networks (WPANS)* (2011).

Performance of AODV Routing Protocol using Group and Entity Mobility Models in Wireless Sensor Networks, *Proceedings of the International Multiconference of Engineers*

Protocol, *Proceedings of the 7th IFAC International Conference on Fieldbuses and Networks*

(ETX) for Wireless Mesh Networks, *Proceedings of the 3rd International Conference on*

ZigBee-based Wireless Sensor Networks, *Proceedings of the 16th IEEE Symposium on*

in wireless sensor networks for health communication systems, *Proceedings of the 2nd International Conference on PErvasive Technologies Related to Assistive Environments*,

provide priority in the contention period to those nodes with better conditions.

allowing multidisciplinary approaches and performance improvements.

*on Mobile Computing and Networking (MobiCom)*.

**8. References**

*(Amby-Sys)*.

*Conputing*.

*and Computer Scientists*.

PETRA, pp. 34:1–34:8.

*Maxfor Technology INC. http://http://www.maxfor.co.kr* (2011).

*in Industrial and Embedded Systems*.

*Computer and Communications (ISCC)*.

Fig. 14. Route length (number of hops).

receive a non-optimal route first and later the optimal one. Note that for small networks (50 nodes or less), the average number of hops is similar for both proposals, but when the network size increases, so does the number of alternative routes, and the probability of receiving a non-optimal route first in AODV increases. This fact can be a disadvantage for networks with real-time requirements due to as source nodes will either have to wait and see if a better RREP is received, or send data using a route that can be non-optimum. As for AODV-ETX, it obtains higher route lengths due to it selects paths not considering the number of hops, but the expected transmissions.

All these results show that AODV-FL is more effective than the original AODV, and the ETX-based approach in all the experiments, reducing the energy consumption by up to 70%. The performance of the route discovery has also been improved, not only in the number of packets (around 60-70% reduction), but also in the path lengths (20% reduction) and end-to-end delay (40-50% reduction).

#### **7. Conclusions and future research**

Monitoring applications in wireless sensor networks require effective, robust and scalable routing protocols, above all in applications with resource-constrained nodes. This chapter details the use of fuzzy logic to improve the routing protocol used by the ZigBee standard in mesh networks, AODV. The use of fuzzy logic as a metric in network routing improves the performance of real networks. AODV-FL uses this metric, achieving an energy reduction of 70% in network route creation, due to a considerable reduction in the number of RREQs generated, reducing collisions and the end-to-end delay. In contrast with other proposals that require additional memory or processing costs, the use of fuzzy logic does not imply an extra load on the system, and it improves the performance of the intelligent dense monitoring of physical environments.

Experimental comparisons with AODV and AODV-ETX endorse the suitability of AODV-FL for implementation in real wireless sensor networks.

Future research can be oriented to the addition of new parameters to the fuzzy logic system, studying the performance achieved by these new variables, such as the number of child nodes, or node density. The use of fuzzy logic in other layers, such as the MAC layer, will help to provide priority in the contention period to those nodes with better conditions.

In summary, fuzzy logic is a powerful approach that has demonstrated to be effective when combining with other disciplines such as routing approaches for WSNs. The potential of fuzzy logic goes beyond traditional control systems and can be used on many research fields, allowing multidisciplinary approaches and performance improvements.

#### **8. References**

18 Will-be-set-by-IN-TECH

0 50 100 150 200

receive a non-optimal route first and later the optimal one. Note that for small networks (50 nodes or less), the average number of hops is similar for both proposals, but when the network size increases, so does the number of alternative routes, and the probability of receiving a non-optimal route first in AODV increases. This fact can be a disadvantage for networks with real-time requirements due to as source nodes will either have to wait and see if a better RREP is received, or send data using a route that can be non-optimum. As for AODV-ETX, it obtains higher route lengths due to it selects paths not considering the number of hops, but

All these results show that AODV-FL is more effective than the original AODV, and the ETX-based approach in all the experiments, reducing the energy consumption by up to 70%. The performance of the route discovery has also been improved, not only in the number of packets (around 60-70% reduction), but also in the path lengths (20% reduction) and

Monitoring applications in wireless sensor networks require effective, robust and scalable routing protocols, above all in applications with resource-constrained nodes. This chapter details the use of fuzzy logic to improve the routing protocol used by the ZigBee standard in mesh networks, AODV. The use of fuzzy logic as a metric in network routing improves the performance of real networks. AODV-FL uses this metric, achieving an energy reduction of 70% in network route creation, due to a considerable reduction in the number of RREQs generated, reducing collisions and the end-to-end delay. In contrast with other proposals that require additional memory or processing costs, the use of fuzzy logic does not imply an extra load on the system, and it improves the performance of the intelligent dense monitoring of

Experimental comparisons with AODV and AODV-ETX endorse the suitability of AODV-FL

Future research can be oriented to the addition of new parameters to the fuzzy logic system, studying the performance achieved by these new variables, such as the number of child nodes,

Number of nodes

1

1.5

2

Hops

Fig. 14. Route length (number of hops).

the expected transmissions.

physical environments.

end-to-end delay (40-50% reduction).

**7. Conclusions and future research**

for implementation in real wireless sensor networks.

2.5

3

AODV AODV−FL AODV−ETX


**12** 

*Brazil* 

**Fuzzy Logic on a Polygenic Multi-Agent** 

Samuel Azevedo, Rummenigge Rudson

*Universidade Federal do Rio Grande do Norte, DCA-CT-UFRN, Campus Universit[ario,* 

and Luiz Gonçalves

 *Lagoa Nova, Natal, RN,* 

**System for Steganalysis of Digital Images** 

Digital cryptography has being a solution for protecting transmission of data in applications such as electronic commerce (Luciano 2003), electronic vote (Kofler 2003), and digital Television (Macq 1995). However, an interceptor monitoring network flow could easily break purely encoded data and clear the contents of cryptographed messages. Steganography techniques came up in order to help improving this protection. The goal of steganography is to hide data into a covering message (envelop) in such a way that an interceptor has no way to notice the presence of a hidden message in its covering envelop. Note that one can combine both cryptography and steganography in order to achieve better security. For example an image can be enriched with visually imperceptible extra information that, when eventually noticed, could be understood as an eventual noise. This damaged image could serve thus as a camouflaging body that brings protected data to the other side of the communication process. Any media object can be used as the covering message, such as text, audio, video, network packages, and file systems. Digital images are known to be the most used media objects for this purpose due to its inherent artistic appeal. In steganography, specifically, the carrying message is a digital object (image, audio, video etc) that envelops hidden data. When a potential covering object carries hidden data it can be called a steganographed object. In order to extract hidden information from this object, one has to know that a conspicuous object is steganographed, what is the steganographic algorithm

used to hide data, and the password that will be generally requested by the algorithm.

contents, for example breaking of privacy is important in criminal investigations.

On the other side, if one would like to reveal the data which is hidden, it should use steganalysis techniques in order to detect whether a message has hidden data or not. This is just the subject approached in this work. Although it is undeniable that everyone has the right to protect some information there are some situations where it is necessary to reveal its

Besides detecting the presence of hidden messages, a useful steganalysis technique should also estimate the length of the messages and also somehow possibly to detect which steganographic algorithm is used to hide information. Since one knows the algorithm to

**1. Introduction** 


### **Fuzzy Logic on a Polygenic Multi-Agent System for Steganalysis of Digital Images**

Samuel Azevedo, Rummenigge Rudson and Luiz Gonçalves *Universidade Federal do Rio Grande do Norte, DCA-CT-UFRN, Campus Universit[ario, Lagoa Nova, Natal, RN, Brazil* 

#### **1. Introduction**

20 Will-be-set-by-IN-TECH

240 Fuzzy Logic – Emerging Technologies and Applications

Pirzada, A. A. & et al. (2007). High performance AODV routing protocol for hybrid wireless

*and Ubiquitous Systems: Computing, Networking and Services (MobiQuitous)*. Ramachandran, K. N., Buddhikot, M. M., Chandranmenon, G., Miller, S., Belding-Royer, E. M.

Sklyarenko, G. (2006). AODV Routing Protocol. Seminar Technische Informatik. Institute für

Vasseur, J. P. (2010). Terminology in Low power And Lossy Networks. Internet

Wang, T. M., Liao, I. J., Liao, J. C., Suen, T. W. & Lee, W. T. (2009). An Intelligent Fuzzy

Yang, Z. & Mohammed, A. (2010). A survey of routing protocols of wireless sensor networks,

Yick, J., Mukherjee, B. & Ghosal, D. (2008). Wireless Sensor Network Survey, *Computer*

Zvikhachevskaya, A. & Mihaylova, L. (2009). Self-organisation in wireless sensor networks

Z. Shelby and C. Bormann (2009). 6LoWPAN: The Wireless Embedded Internet. Wiley. Zhao, F. & Guibas, L. (2004). *Wireless Sensor Networks, an Information Processing Approach*,

for assisted living, *Proceedings of the IET Assisted Living Conference*.

draft, Networking Working Group. http://tools.ietf.org/html/draft-ietf-roll

Controller for Air-Condition with ZigBee Sensors, *International Journal in Smart*

Design Paradigms, *IEEE Personal Communications* 6(2): 46–55.

*Proceedings of the Sustainable Wireless Sensor Networks*.

*ZigBee Specification, ZigBee Alliance* (2011). http://www.zigbee.org/.

Informatik, Freie Universität Berlin.

*Sensong and Intelligent Systems* 2. *Wisevine project, http://www.wisevine.info/* (2011).


*Networks* 52.

Elsevier.

mesh networks , *Proceedings of The Fourth Annual International Conference on Mobile*

& Almeroth, K. C. (2005). On the Design and Implementation of Infraestructure Mesh Networks, *Proceedings of the IEEE Workshop on Wireless Mesh Networks (WiMesh)*. Ran, G., Zhang, H. & Gong, S. (2010). Improving on LEACH Protocol of Wireless Sensor Networks Using Fuzzy-Logic, *Journal of Information and Computational Science* 7(3). Royer, E. & Toh, C. K. (1999). Self-Organization in Communication Networks: Principles and

> Digital cryptography has being a solution for protecting transmission of data in applications such as electronic commerce (Luciano 2003), electronic vote (Kofler 2003), and digital Television (Macq 1995). However, an interceptor monitoring network flow could easily break purely encoded data and clear the contents of cryptographed messages. Steganography techniques came up in order to help improving this protection. The goal of steganography is to hide data into a covering message (envelop) in such a way that an interceptor has no way to notice the presence of a hidden message in its covering envelop. Note that one can combine both cryptography and steganography in order to achieve better security. For example an image can be enriched with visually imperceptible extra information that, when eventually noticed, could be understood as an eventual noise. This damaged image could serve thus as a camouflaging body that brings protected data to the other side of the communication process. Any media object can be used as the covering message, such as text, audio, video, network packages, and file systems. Digital images are known to be the most used media objects for this purpose due to its inherent artistic appeal.

> In steganography, specifically, the carrying message is a digital object (image, audio, video etc) that envelops hidden data. When a potential covering object carries hidden data it can be called a steganographed object. In order to extract hidden information from this object, one has to know that a conspicuous object is steganographed, what is the steganographic algorithm used to hide data, and the password that will be generally requested by the algorithm.

> On the other side, if one would like to reveal the data which is hidden, it should use steganalysis techniques in order to detect whether a message has hidden data or not. This is just the subject approached in this work. Although it is undeniable that everyone has the right to protect some information there are some situations where it is necessary to reveal its contents, for example breaking of privacy is important in criminal investigations.

> Besides detecting the presence of hidden messages, a useful steganalysis technique should also estimate the length of the messages and also somehow possibly to detect which steganographic algorithm is used to hide information. Since one knows the algorithm to

Fuzzy Logic on a Polygenic Multi-Agent System for Steganalysis of Digital Images 243

 "Machine Learning is the AI field which aims to develop computational techniques about learning as well as the construction of systems capable to acquire knowledge in an

 *Symbolic paradigm* – builds a symbolic representation of the problems´ solution through the analysis of examples, the machine learning most known methods of this paradigm

 *Statistical paradigm* – composed by the classification methods that try to analyze statistics in order to find an statistical model approximated to the problem; a known

 *Paradigm based in examples* – classifies one instance (or sample) through its comparison with other previously classified samples, returning as result the class of the classified instance that is more similar to it; the most known method of this paradigm is the K-nn which returns the class that appears the most in the *k* nearest neighbors to a consulted

 *Connectionist paradigm* – based upon the biological metaphor of neural connections of the nervous system, it try to train a network of neurons with samples in a way that the

Every learning method presents, after training, an error or accuracy rate. Other important rates are the True Positive and True Negative rates that indicate respectively the rates of positive and negative cases correctly detected. Many times, one wish to improve these rates and one of the improving strategies are the ensembles or clustering of classifiers. In steganalysis, a critical rate is the False Negative, which indicates the percent of cases that

Agents are autonomous software entities that act in a certain environment and are capable of taking decisions as which actions to perform in order to reach any goal (Russel 1995).

A multi-agent system is a complex system in which several specialized or redundant agents interact, cooperating, negotiating, and exchanging information in order to reach any optimal

MAS are systems that contain a set of software agents working together that interact between them and with the environment through some communication channel. Agents have areas of influence in the environment that may or not overlap (Wooldridge 2001). They

weights of its connections are adjusted to solve the problem of classification. *Evolutionary or genetic paradigm* – this is also based in biological metaphor, in this case the genetic evolution; it consists in realizing crossings and mutations in a set of classifiers to solve a problem; during N interactions (or generations), the classifiers with best performance in each generation prevail and the next generation of classifiers is generated by variations of these; the genes are the parameters of the classifiers, that can be of any of the other paradigms, but instead of regular training to accurate they

Among the machine learning paradigms, we have (Sanches 2004):

method of this paradigm is the *Bayesian Learning* algorithm.

parameters, these parameters are changed through evolution.

were incorrectly classified regular images but in fact contained hidden data.

**2.1 Machine learning** 

sample.

**2.2 Multi-agent systems** 

goal.

automatic way." (Rezende 2003)

are *decision trees* and *semantic networks*.

hide or to reveal the message contents, and the steganographed object, one may try some common known attacks to break the password like the brute force password guessing.

When new cryptographic or steganographic techniques arise, new cryptoanalysis or steganalysis also are developed addressing the new characteristics of the problem. So, one can say that there is a race between cryptographers and cryptoanalyzers and between steganographers and steganalysers. A technological advance in one side forces the other to overcome it.

Other characteristic of the problem is that when new steganalysis techniques are developed, new steganographic techniques arise immune to the existent attacks. Therefore steganalysis systems demand flexibility to adapt to the new steganographies. This flexibility can be obtained by learning or by using software engineering techniques that ease the alteration of the system in a handful time (such as modularization, documentation, etc).

In this work we approach steganalysis for digital images, which represent a vast distribution of data around the Internet. Due to the very complex nature of the problem, it is generally required to perform steganalysis on a huge volume of data. Of course it would be adequate to perform this in an autonomously way by using a computational system. Autonomy and flexibility are characteristics present in software entities called agents. By the complexity of the problem, these agents would be more appropriately approached in a Multi-Agent System (MAS), which is a system where several specialized or redundant agents interact (through cooperation, negotiation, and exchanging information, for example) to achieve their goals.

Since MAS are systems that approach social interaction between agents, we need to model the way these interactions will be performed. It is common to use metaphors from nature as heuristics in order to solve computational problems in a less complex way. A good heuristic for this solution would be inspired in social interaction of insect communities. Social insects present important characteristics of MAS such as cooperation, distribution of multiple tasks, and coordination. Our work is inspired in the polygenic societies of bees from the species Melipona Bicolor where several queens of a hive can cooperate in the coordination of all the workers. We initially apply such coordination model to our MAS, where each worker is a classifier, and further apply fuzzy logic to solve the classification of heterogeneous classifiers to a same sample.

Therefore, our proposal and main contribution is a multi-agent system for digital image steganalysis that is based on the paradigm of the community of polygenic bees using fuzzy logic. With such approach we aim to solve the problem of automatic steganalysis for digital media with a case study on digital images. The architecture proposed here is designed to detect if a file is suspicious of carrying hidden contents allowing to attempt to extract them with other techniques (such as brute force password guessing). Experimental results validate the system, showing the applicability of the MAS to steganalysis of image data.

#### **2. Background and methods**

In order to better understand our problem, some background must be addressed in different areas of knowledge including cryptology, machine learning, MAS, heuristics, image segmentation, and fuzzy logic.

#### **2.1 Machine learning**

242 Fuzzy Logic – Emerging Technologies and Applications

hide or to reveal the message contents, and the steganographed object, one may try some common known attacks to break the password like the brute force password guessing.

When new cryptographic or steganographic techniques arise, new cryptoanalysis or steganalysis also are developed addressing the new characteristics of the problem. So, one can say that there is a race between cryptographers and cryptoanalyzers and between steganographers and steganalysers. A technological advance in one side forces the other to

Other characteristic of the problem is that when new steganalysis techniques are developed, new steganographic techniques arise immune to the existent attacks. Therefore steganalysis systems demand flexibility to adapt to the new steganographies. This flexibility can be obtained by learning or by using software engineering techniques that ease the alteration of

In this work we approach steganalysis for digital images, which represent a vast distribution of data around the Internet. Due to the very complex nature of the problem, it is generally required to perform steganalysis on a huge volume of data. Of course it would be adequate to perform this in an autonomously way by using a computational system. Autonomy and flexibility are characteristics present in software entities called agents. By the complexity of the problem, these agents would be more appropriately approached in a Multi-Agent System (MAS), which is a system where several specialized or redundant agents interact (through cooperation, negotiation, and exchanging information, for example) to achieve

Since MAS are systems that approach social interaction between agents, we need to model the way these interactions will be performed. It is common to use metaphors from nature as heuristics in order to solve computational problems in a less complex way. A good heuristic for this solution would be inspired in social interaction of insect communities. Social insects present important characteristics of MAS such as cooperation, distribution of multiple tasks, and coordination. Our work is inspired in the polygenic societies of bees from the species Melipona Bicolor where several queens of a hive can cooperate in the coordination of all the workers. We initially apply such coordination model to our MAS, where each worker is a classifier, and further apply fuzzy logic to solve the classification of heterogeneous

Therefore, our proposal and main contribution is a multi-agent system for digital image steganalysis that is based on the paradigm of the community of polygenic bees using fuzzy logic. With such approach we aim to solve the problem of automatic steganalysis for digital media with a case study on digital images. The architecture proposed here is designed to detect if a file is suspicious of carrying hidden contents allowing to attempt to extract them with other techniques (such as brute force password guessing). Experimental results validate the system, showing the applicability of the MAS to steganalysis of image data.

In order to better understand our problem, some background must be addressed in different areas of knowledge including cryptology, machine learning, MAS, heuristics, image

the system in a handful time (such as modularization, documentation, etc).

overcome it.

their goals.

classifiers to a same sample.

**2. Background and methods** 

segmentation, and fuzzy logic.

 "Machine Learning is the AI field which aims to develop computational techniques about learning as well as the construction of systems capable to acquire knowledge in an automatic way." (Rezende 2003)

Among the machine learning paradigms, we have (Sanches 2004):


Every learning method presents, after training, an error or accuracy rate. Other important rates are the True Positive and True Negative rates that indicate respectively the rates of positive and negative cases correctly detected. Many times, one wish to improve these rates and one of the improving strategies are the ensembles or clustering of classifiers. In steganalysis, a critical rate is the False Negative, which indicates the percent of cases that were incorrectly classified regular images but in fact contained hidden data.

#### **2.2 Multi-agent systems**

Agents are autonomous software entities that act in a certain environment and are capable of taking decisions as which actions to perform in order to reach any goal (Russel 1995).

A multi-agent system is a complex system in which several specialized or redundant agents interact, cooperating, negotiating, and exchanging information in order to reach any optimal goal.

MAS are systems that contain a set of software agents working together that interact between them and with the environment through some communication channel. Agents have areas of influence in the environment that may or not overlap (Wooldridge 2001). They

Fuzzy Logic on a Polygenic Multi-Agent System for Steganalysis of Digital Images 245

Methods based in edge detection, histogram statistics and clustering, and transform domain error prediction, are found in many of the current solutions for steganalysis, as the

In our work, we use some of the methods above to compose the features that compose an instance for the machine learning algorithms. These features use statistical information such as mean, variance, asymmetry and kurtosis. Mean is the first statistical momentum, variance, asymmetry and kurtosis are, respectively the second, third and fourth momentums over the mean. The equation bellow shows the general formula to find the kth momentum (the mean is the first momentum, but its value is 0). The equation also can be

*k k*

( ) ( ) () . *xx x <sup>f</sup> x dx*

Since the publication of "Fuzzy sets" (Zadeh 1965), many studies have been done to apply fuzzy logic in diverse fields. In machine learning, fuzzy logic has been applied to algorithms from different paradigms as well as to ensembles of classifiers, for example: Support Vector Machines (Lin 2002); neural networks (Carpenter 1992), (Jang 1993); and decision trees (Acampora 2011). López-Ortega (2011) points out that fuzzy clustering and MAS lead to

In software agency, we can see the use of fuzzy knowledge based systems (Arroyo 2011) to implement the decision making process and actions of software agents. Also, we can observe

In steganalysis, the most common use of fuzzy logic is presented in the use of fuzzy machine learning algorithms and in fuzzy clustering (see the related work in 2.6 for further details). One of the main contributions of this work is the design of a novel fuzzy clustering

Steganography is a subarea of information security that includes several other inner areas meaning covert written (Katzenbeisser 2000). In general, its focus is the inclusion of information in a media data that is not suspect. In fact, it is the art of occluding data in data (Artz 2001). When two communication sides A and B want to exchange a secret message, they use an occulting message (or covering object, envelop, mule) applying some steganography technique that may use or not some key *k* obtaining in this way a steganographic message that is undistinguishable from the previous. This last is sent through the communication channel. There are several techniques for doing steganalysis as: 1. S**ubstitution system** – redundant parts of the media are substituted by the data that one

2. T**echniques in the transform domain** – insert secret data in the signal transform

3. S**pecter scattering** – the specter of distribution of the information is scattered;

the use of fuzzy logic theory for agents coordination (Goodarzi 2011), (Hagras 2010).

 

read as *E[(X − E[X])k]* where *X* is a random variable, and *E[X]* is the expected value.

*k* 

discussed in section 2.6.

**2.5 Fuzzy logic** 

high quality decisions.

**2.6 Cryptology** 

wants to occult;

domain (frequency domain);

approach using coordination of agents.

can interact through the use of negotiation, coordination, or cooperation. Bid, argumentation or game theory can also be used by agents (Macedo 2001).

A society of agents may be composed by homogeneous or heterogeneous agents. The coordination problem is how to manage the interdependencies of tasks and resources between agents. Wooldridge classifies four models of coordination: global-partial planning, joint intentions, mutual modeling, and social rules.

In our problem, we use a heuristic of social bees to coordinate the collective work of agents, and the we approach in this MAS a fuzzy clustering algorithm to enhance the detection of hidden data into images.

Fig. 1. M. Bicolor queens in reproduction process. Two or more bees can put much more eggs thus diminishing a lot the efforts for getting a mature colony.

#### **2.3 Heuristics**

Heuristics can be devised base on approaches as genetic algorithm, memetic algorithms, simulated annealing and insect colonies as ant and bees. Algorithms that use metaphors based on colonies aim to imitate some behavior of those in order to search solutions for complex problems. Biologically, social insects may be monogenic or polygenic. That means, it can exist societies that present a single or several queens at the same time (Aponte 2003). Bees of the specie Meliponine Bicolor (see Figure 1) can be polygenic.

#### **2.4 Image segmentation**

Since steganography aims to hide the existence of data within data, it´s important to find computer vision techniques that are able to see this hidden information in images. The most simple steganographic algorithms aim to hide data in the less significant bit of each pixel, these generally are imperceptive to human eye, but they generate distortions in images easily detected by common image segmentation algorithms.

Although there are general purpose techniques and algorithms for image segmentation, they often must be combined with domain knowledge to effectively solve a vision problem; thus, image segmentation must be approached by many perspectives (Pavidlis 1982).

Methods based in edge detection, histogram statistics and clustering, and transform domain error prediction, are found in many of the current solutions for steganalysis, as the discussed in section 2.6.

In our work, we use some of the methods above to compose the features that compose an instance for the machine learning algorithms. These features use statistical information such as mean, variance, asymmetry and kurtosis. Mean is the first statistical momentum, variance, asymmetry and kurtosis are, respectively the second, third and fourth momentums over the mean. The equation bellow shows the general formula to find the kth momentum (the mean is the first momentum, but its value is 0). The equation also can be read as *E[(X − E[X])k]* where *X* is a random variable, and *E[X]* is the expected value.

$$
\mu\_k = \left\langle \left( \mathfrak{x} - \left\langle \mathfrak{x} \right\rangle \right)^k \right\rangle = \int\_{-\infty}^{+\infty} \left( \mathfrak{x} - \mu \right)^k f(\mathfrak{x}) d\mathfrak{x} \dots
$$

#### **2.5 Fuzzy logic**

244 Fuzzy Logic – Emerging Technologies and Applications

can interact through the use of negotiation, coordination, or cooperation. Bid, argumentation

A society of agents may be composed by homogeneous or heterogeneous agents. The coordination problem is how to manage the interdependencies of tasks and resources between agents. Wooldridge classifies four models of coordination: global-partial planning,

In our problem, we use a heuristic of social bees to coordinate the collective work of agents, and the we approach in this MAS a fuzzy clustering algorithm to enhance the detection of

Fig. 1. M. Bicolor queens in reproduction process. Two or more bees can put much more

Heuristics can be devised base on approaches as genetic algorithm, memetic algorithms, simulated annealing and insect colonies as ant and bees. Algorithms that use metaphors based on colonies aim to imitate some behavior of those in order to search solutions for complex problems. Biologically, social insects may be monogenic or polygenic. That means, it can exist societies that present a single or several queens at the same time (Aponte 2003).

Since steganography aims to hide the existence of data within data, it´s important to find computer vision techniques that are able to see this hidden information in images. The most simple steganographic algorithms aim to hide data in the less significant bit of each pixel, these generally are imperceptive to human eye, but they generate distortions in images

Although there are general purpose techniques and algorithms for image segmentation, they often must be combined with domain knowledge to effectively solve a vision problem;

thus, image segmentation must be approached by many perspectives (Pavidlis 1982).

eggs thus diminishing a lot the efforts for getting a mature colony.

Bees of the specie Meliponine Bicolor (see Figure 1) can be polygenic.

easily detected by common image segmentation algorithms.

or game theory can also be used by agents (Macedo 2001).

joint intentions, mutual modeling, and social rules.

hidden data into images.

**2.3 Heuristics** 

**2.4 Image segmentation** 

Since the publication of "Fuzzy sets" (Zadeh 1965), many studies have been done to apply fuzzy logic in diverse fields. In machine learning, fuzzy logic has been applied to algorithms from different paradigms as well as to ensembles of classifiers, for example: Support Vector Machines (Lin 2002); neural networks (Carpenter 1992), (Jang 1993); and decision trees (Acampora 2011). López-Ortega (2011) points out that fuzzy clustering and MAS lead to high quality decisions.

In software agency, we can see the use of fuzzy knowledge based systems (Arroyo 2011) to implement the decision making process and actions of software agents. Also, we can observe the use of fuzzy logic theory for agents coordination (Goodarzi 2011), (Hagras 2010).

In steganalysis, the most common use of fuzzy logic is presented in the use of fuzzy machine learning algorithms and in fuzzy clustering (see the related work in 2.6 for further details). One of the main contributions of this work is the design of a novel fuzzy clustering approach using coordination of agents.

#### **2.6 Cryptology**

Steganography is a subarea of information security that includes several other inner areas meaning covert written (Katzenbeisser 2000). In general, its focus is the inclusion of information in a media data that is not suspect. In fact, it is the art of occluding data in data (Artz 2001). When two communication sides A and B want to exchange a secret message, they use an occulting message (or covering object, envelop, mule) applying some steganography technique that may use or not some key *k* obtaining in this way a steganographic message that is undistinguishable from the previous. This last is sent through the communication channel. There are several techniques for doing steganalysis as:


Fuzzy Logic on a Polygenic Multi-Agent System for Steganalysis of Digital Images 247

3. **Features Calibration** - one must select the features used to describe the data that will be applied to the machine learning technique; in the case of images, these features may be statistical data from histogram or from segments of the image, errors found in predicted coefficient values in the transform domain, and so on. This can be achieved by choosing an initial set of features (by literature, empirical experience, experimentation, etc) and testing subsets of these features in the next phases to verify the optimal subset of features. Liu (2008b) describes an interesting methodology for feature mining for

4. **Data Samples** - it´s necessary to create a database with samples fitting the features selected, and this database should able to be accessed by the classifier. But two random subsets of samples might be separated, the bigger to the training and smaller to the testing phases. The size of these databases is another issue, the ideal, statistically speaking is that this size should be big enough to represent the population of real cases; but by the nature of the problem, there are no statistics describing how many stego objects are there in the world; so, there are works using from 30 to more than 30000 samples. It´s important to say that the samples must be in quantities proportional to the different classes (from non-stego objects/stego objects in the most simple cases; to non-

stego objects/stego object for algorithm 1…N in most complex solutions).

the accuracy, true positive and true negatives rates must be calculated.

5. **Training** - this phase is about training the machine learning mechanisms with the training subset of data, according to the algorithm selected, this step may last a long

6. **Testing** - finally, after trained, the classifier may be submitted to the testing dataset, and

Bakhshandeh (2009) presents a steganalysis technique based on local information and human visual system. By performing segmentation and analysis for clustering these segments, the best segments are chosen for steganalysis. The algorithm they have used for classification is Fuzzy Clustering, simplifying, one may say that they give a fuzzy weight to the results of many classifiers, and use a clustering algorithm to decide the final classification results. Wavelet information is extracted to compose the feature used in a SVM algorithm to classify whether an image has or does not have hidden data. The results are at first sight promising, but if one consider that their experiments were in images carrying hidden data in 100% of theirs spectrum capability for spread spectrum steganography, one would expect to see the results for messages that are smaller the full capability of the cover image, since it´s more difficult to detect the presence of smaller data because the resulting

In the work of Liu and Sung (Liu 2008) it is presented a steganalysis technique that uses One-Against-All decomposition for SVM to classify whether or not a jpeg image contains hidden data in one of three steganographic techniques, based in detecting errors from predicted DFT, DCT or DWT coefficients. After this classification, they use a Dynamical Evolutionary Neuro-Fuzzy Inference Systems (DENFIS), to estimate the length of the hidden

calibration and obtaining data samples, or after these steps.

steganalysis.

time.

alteration is smaller in the cover image data.

used, also a clustering technique must be defined to combine the classification results of different algorithms. Sometimes, the architecture of the final algorithm must be redesigned and applied to new trainings and tests in order find improved results. According to the complexity of the features and data that will be analyzed, one may choose a most fitting machine learning solution. This can be performed before features


An important characteristic in steganography is determining the capacity of an object to hide information, we can observe this concept in what Moskowitz (2002) calls Capability:

*"Capability = (P;D) where P is the payload size and D is a detectability threshold. We sometimes expand the capability to a triple (P; D; R) where R is a measure of robustness of the stego channel."* 

The quoted author also states there for steganography in the least significant bit of images, the payload is limited from 0 to 50% of the size of the carry image, otherwise changing the cover to a negative.

Steganalysis goal is to attack or monitor a communication channel in order to detect existing information that is occulted in messages or to forge some occult message, interrupt communication, and to extract occulted data.

Different approaches to steganalysis can be found in the literature as visual attack (Fridrich, 2002, 2004), statistical analysis (Katzenbeisser 2000), and signature detection (Chandramouli 2004). The first approaches the most elementary methods, as for example the bit substitution systems that may cause visible distortions to images, what reveals the existence of hidden contents. Statistical analysis looks statistical measures in files as the histogram to verify common aberrations. It can use pure statistic methods or some combination with machine learning. In signature detection approaches, any degradation caused by steganography methods can be read as a signature of these methods. These methods generally span suspect files to find signatures in the data noise that can reveal if any steganography approach is used including some times which was the used approach.

There are two categories of steganalysis techniques: specific and universal (or blind) steganalysis. While specific steganalysis is related to attack objects generated by one single steganographic algorithm, universal steganalysis aim to attack stego objects independent of the steganographic algorithm used. Commonly, steganalysis use machine learning algorithms in order to classify whether an object may contain hidden data or not.

In order to create a steganalysis algorithm, one must think in six phases or steps:


4. **Statistical methods** – produces steganographed data through statistical manipulation

5. **Distortion techniques** – produces distortions in a covering media in order to get steganographed data, compares the original covering media with the modified in order

An important characteristic in steganography is determining the capacity of an object to hide information, we can observe this concept in what Moskowitz (2002) calls Capability: *"Capability = (P;D) where P is the payload size and D is a detectability threshold. We sometimes expand the capability to a triple (P; D; R) where R is a measure of robustness of the stego channel."*  The quoted author also states there for steganography in the least significant bit of images, the payload is limited from 0 to 50% of the size of the carry image, otherwise changing the

Steganalysis goal is to attack or monitor a communication channel in order to detect existing information that is occulted in messages or to forge some occult message, interrupt

Different approaches to steganalysis can be found in the literature as visual attack (Fridrich, 2002, 2004), statistical analysis (Katzenbeisser 2000), and signature detection (Chandramouli 2004). The first approaches the most elementary methods, as for example the bit substitution systems that may cause visible distortions to images, what reveals the existence of hidden contents. Statistical analysis looks statistical measures in files as the histogram to verify common aberrations. It can use pure statistic methods or some combination with machine learning. In signature detection approaches, any degradation caused by steganography methods can be read as a signature of these methods. These methods generally span suspect files to find signatures in the data noise that can reveal if any steganography approach is

There are two categories of steganalysis techniques: specific and universal (or blind) steganalysis. While specific steganalysis is related to attack objects generated by one single steganographic algorithm, universal steganalysis aim to attack stego objects independent of the steganographic algorithm used. Commonly, steganalysis use machine learning

1. **Steganalysis goals** - Consists in defining and implementing the category of technique will be performed (specific or universal), and defining which attacks will be realized, as detection of hidden information, data estimation (as for example the length of the hidden data), steganographic algorithm used. The following types of attack don´t need a classifier, and so, if they are isolated attacks it´s not needed to implement the steps 2 to 6, but if combined with the other attacks mentioned above, these steps are still necessary: data extraction (as password guessing from a dictionary), intercept the cover messages (such as sniffing network packages), denial of service (applying noise to an image, disabling the possibility to extract hidden content), and forging a hidden

2. **Classifier Method**- one must choose and implement which machine learning algorithms will be used to the classification process. If more than one classifier will be

algorithms in order to classify whether an object may contain hidden data or not. In order to create a steganalysis algorithm, one must think in six phases or steps:

of covering data;

to extract them.

cover to a negative.

communication, and to extract occulted data.

used including some times which was the used approach.

message to confound the communication.

used, also a clustering technique must be defined to combine the classification results of different algorithms. Sometimes, the architecture of the final algorithm must be redesigned and applied to new trainings and tests in order find improved results. According to the complexity of the features and data that will be analyzed, one may choose a most fitting machine learning solution. This can be performed before features calibration and obtaining data samples, or after these steps.


Bakhshandeh (2009) presents a steganalysis technique based on local information and human visual system. By performing segmentation and analysis for clustering these segments, the best segments are chosen for steganalysis. The algorithm they have used for classification is Fuzzy Clustering, simplifying, one may say that they give a fuzzy weight to the results of many classifiers, and use a clustering algorithm to decide the final classification results. Wavelet information is extracted to compose the feature used in a SVM algorithm to classify whether an image has or does not have hidden data. The results are at first sight promising, but if one consider that their experiments were in images carrying hidden data in 100% of theirs spectrum capability for spread spectrum steganography, one would expect to see the results for messages that are smaller the full capability of the cover image, since it´s more difficult to detect the presence of smaller data because the resulting alteration is smaller in the cover image data.

In the work of Liu and Sung (Liu 2008) it is presented a steganalysis technique that uses One-Against-All decomposition for SVM to classify whether or not a jpeg image contains hidden data in one of three steganographic techniques, based in detecting errors from predicted DFT, DCT or DWT coefficients. After this classification, they use a Dynamical Evolutionary Neuro-Fuzzy Inference Systems (DENFIS), to estimate the length of the hidden

Fuzzy Logic on a Polygenic Multi-Agent System for Steganalysis of Digital Images 249

 When the attributed workers realize the classification, the queen uses a fuzzy inference system to combine the results of these workers according to their fuzzy weights and

When more than one coordination agent come to divergent results, they start a negotiation

The specialized and the general classifier agents represent different specialized workers in this metaphor. The communication between the agents is realized through a message board. The accuracy rate for the general classifier agents and the specialized classifier agents can be fuzzyficated as shown in the graphic bellow. Where L, M and H means Low, Medium and High accurate rates, respectively (Figure 3.a). In order to linearize the classification problem, the classification process will give a probability a given sample is or not a stego object according to the classification (Figure 3.b). The inference method used is a simple Mamdani

Fig. 3. Fuzzyfication. a) accuracy rates; b) probability of a classified sample being stego

50% 100% Accuracy

Figure 5 presents the use case diagram of the approach. The use cases presented are: monitoring of files, negotiation of final result, coordination of classification, attributing/instantiating agents, and classification; and these actions are realized as described next. The actors that are present at the use case of the system are the monitors (monitor agent),

50% 100%

L M H

finds her classification result.

process in order to find the final classification result.

Fig. 2. Polygenic MAS Fuzzy Clustering Steganalysis Architecture.

FIS, and the Figure 4 simplifies its mechanisms.

Stego

object.

class

**3.2 Use case view** 

Non-stego

data. The estimative found was very accurate for F5 steganographic algorithm, but not so effective for others.

Amirkhani (2011) highlights that blind steganalysis algorithms use to have each internal similar (or a same) processes for different image categories (smooth, complex, noisy, etc), instead of using the particular characteristics of an image type to attack it. Their framework can make use of any steganalysis technique that are applied to two main modifications: before training, the images must be divided into different content classes; and the result of a classifier must be weighted to a fuzzy value according to the content class trained, after that, the result is combined in order to classify if an image is a regular image or has a hidden content, these two final classes are called by the authors of Cover (regular) and Stego (has hidden content). They experiment this framework with some known steganalysis algorithms and confront their efficacy with several steganographic algorithms, showing discrete increases of accuracy, true positives and true negatives rates. In our approach, we train some of our classifiers for different image types, and other for general image types, in order to further clustering their results.

#### **3. Polygenic MAS fuzzy clustering steganalysis**

The MAS system approached here, according to the taxonomy presented by Rezende (2003), may be classified as a heterogeneous agent open system, with low initial granularity.

The main issue is social resolution that aims to solve the problem of steganalysis in a cooperative and distributed way. However, it is also approached the social simulation view for simulating the behavior of polygenic bees. Interaction patterns present in the system are commensalism (in the interactions between classifier and coordinator agents), and protocooperation, in interactions between classifier agents.

#### **3.1 Architectural view**

Figure 2 presents the general architecture of the proposed solution. The steganalysis process is realized in 2 steps: first, it´s necessary to perform some type of data interception (such as network packets sniffing) – this is not approached in this work; then, the intercepted data may finally be classified with our approach into stego data (data that carries hidden content) or non-stego data (without hidden content).

 The polygenic heuristics here is present in the coordinator agents, which can represent the queens of this society. They are responsible to ask for specialized and general classifiers agents to analyze an intercepted image file. These queens also perform the following fuzzy clustering approach:


 When the attributed workers realize the classification, the queen uses a fuzzy inference system to combine the results of these workers according to their fuzzy weights and finds her classification result.

When more than one coordination agent come to divergent results, they start a negotiation process in order to find the final classification result.

The specialized and the general classifier agents represent different specialized workers in this metaphor. The communication between the agents is realized through a message board.

The accuracy rate for the general classifier agents and the specialized classifier agents can be fuzzyficated as shown in the graphic bellow. Where L, M and H means Low, Medium and High accurate rates, respectively (Figure 3.a). In order to linearize the classification problem, the classification process will give a probability a given sample is or not a stego object according to the classification (Figure 3.b). The inference method used is a simple Mamdani FIS, and the Figure 4 simplifies its mechanisms.

Fig. 3. Fuzzyfication. a) accuracy rates; b) probability of a classified sample being stego object.

#### **3.2 Use case view**

248 Fuzzy Logic – Emerging Technologies and Applications

data. The estimative found was very accurate for F5 steganographic algorithm, but not so

Amirkhani (2011) highlights that blind steganalysis algorithms use to have each internal similar (or a same) processes for different image categories (smooth, complex, noisy, etc), instead of using the particular characteristics of an image type to attack it. Their framework can make use of any steganalysis technique that are applied to two main modifications: before training, the images must be divided into different content classes; and the result of a classifier must be weighted to a fuzzy value according to the content class trained, after that, the result is combined in order to classify if an image is a regular image or has a hidden content, these two final classes are called by the authors of Cover (regular) and Stego (has hidden content). They experiment this framework with some known steganalysis algorithms and confront their efficacy with several steganographic algorithms, showing discrete increases of accuracy, true positives and true negatives rates. In our approach, we train some of our classifiers for different image types, and other for general image types, in order

The MAS system approached here, according to the taxonomy presented by Rezende (2003),

The main issue is social resolution that aims to solve the problem of steganalysis in a cooperative and distributed way. However, it is also approached the social simulation view for simulating the behavior of polygenic bees. Interaction patterns present in the system are commensalism (in the interactions between classifier and coordinator agents), and proto-

Figure 2 presents the general architecture of the proposed solution. The steganalysis process is realized in 2 steps: first, it´s necessary to perform some type of data interception (such as network packets sniffing) – this is not approached in this work; then, the intercepted data may finally be classified with our approach into stego data (data that carries hidden content)

 The polygenic heuristics here is present in the coordinator agents, which can represent the queens of this society. They are responsible to ask for specialized and general classifiers agents to analyze an intercepted image file. These queens also perform the following fuzzy

 According to the training, an **specialized classifier agent** may be more suitable to an image than other, so, it receives a bigger weight when classifying an image which

**general classifier agents** are trained to diverse categories of images, and they receive a

 The **coordinator agent** (or coordination agent) responsible for an specific file asks for specialized and general agents to classify this file; when there is not enough available agents, this queen instantiates new workers of both types and attributes the

may be classified as a heterogeneous agent open system, with low initial granularity.

effective for others.

to further clustering their results.

**3.1 Architectural view** 

clustering approach:

**3. Polygenic MAS fuzzy clustering steganalysis** 

cooperation, in interactions between classifier agents.

or non-stego data (without hidden content).

category it was specialized;

constant weight parameter;

classification for them;

Figure 5 presents the use case diagram of the approach. The use cases presented are: monitoring of files, negotiation of final result, coordination of classification, attributing/instantiating agents, and classification; and these actions are realized as described next. The actors that are present at the use case of the system are the monitors (monitor agent),

Fuzzy Logic on a Polygenic Multi-Agent System for Steganalysis of Digital Images 251

coordination agent, which will use it fuzzy clustering inference algorithm to define the classification result. After the coordination agents find their classification results, if

Fig. 5. Use Case Diagram of Polygenic MAS Fuzzy Clustering Steganalysis.

6. IF TAAR=High and OAAR=Medium and TC!=OC THEN class=TC 7. IF TAAR=High and OAAR=Low and TC!=OC THEN class=TC 8. IF TAAR=Medium and OAAR=High and TC!=OC THEN class=OC

13. IF TAAR=Medium and OAAR=Low and TC!=OC THEN class=TC 14. IF TAAR=Low and OAAR=High and TC!=OC THEN class=OC 15. IF TAAR=Low and OAAR=Medium and TC!=OC THEN class=OC

2. IF TAAR=High and OAAR=High and TC!=OC and TAAR>OAAR THEN

3. IF TAAR=High and OAAR=High and TC!=OC and TAAR=OAAR and TAFN < OAFN

4. IF TAAR=High and OAAR=High and TC!=OC and TAAR=OAAR and TAFN >=

5. IF TAAR=High and OAAR=High and TC!=OC and TAAR<OAAR THEN

9. IF TAAR=Medium and OAAR=Medium and TC!=OC and TAAR>OAAR THEN

10. IF TAAR=Medium and OAAR=Medium and TC!=OC and TAAR=OAAR and TAFN <

11. IF TAAR=Medium and OAAR=Medium and TC!=OC and TAAR=OAAR and TAFN

12. IF TAAR=Medium and OAAR=Medium and TC!=OC and TAAR<OAAR THEN

16. IF TAAR=Low and OAAR=Low and TC!=OC and TAAR>OAAR THEN

All the inference rules of the negotiation protocol follow:

1. IF TC=OC THEN accord

TAAR=TAAR+0.01

TAAR=TAAR-0.01

TAAR=TAAR+0.01

TAAR=TAAR-0.01

TAAR=TAAR+0.01

THEN TAAR=TAAR+0.01

OAFN THEN TAAR=TAAR-0.01

OAFN THEN TAAR=TAAR+0.01

>= OAFN THEN TAAR=TAAR-0.01

divergent, they negotiate to find a final answer.

queens (coordinator agents), and laborers (general and specialized classifier agents). The role of **monitor** can be performed by simple users of the system that submit a set of files to be monitored by the system agents to work on them. Alternatively, one can program monitor agents to perform searchers or sniffs in the internet in order to collect and analyze data.

Fig. 4. Fuzzy inference system for classification of stego images (simplification).

After the monitor agent (or user) perform the monitoring of files, random coordination agents are attributed that file and individually start coordination the global classification process. This process, in his turn, needs the attribution or instantiation of general and specialized classifier agents to the task of classifying the monitored file. When a instantiation is needed, the agents are trained and tested in order to receive a weight corresponding to their adequacy in the classification process. So, the roles of general and specialized classification agents are only responsible to classify the data and send this result to the coordination agent, which will use it fuzzy clustering inference algorithm to define the classification result. After the coordination agents find their classification results, if divergent, they negotiate to find a final answer.

All the inference rules of the negotiation protocol follow:

1. IF TC=OC THEN accord

250 Fuzzy Logic – Emerging Technologies and Applications

queens (coordinator agents), and laborers (general and specialized classifier agents). The role of **monitor** can be performed by simple users of the system that submit a set of files to be monitored by the system agents to work on them. Alternatively, one can program monitor

agents to perform searchers or sniffs in the internet in order to collect and analyze data.

Fig. 4. Fuzzy inference system for classification of stego images (simplification).

After the monitor agent (or user) perform the monitoring of files, random coordination agents are attributed that file and individually start coordination the global classification process. This process, in his turn, needs the attribution or instantiation of general and specialized classifier agents to the task of classifying the monitored file. When a instantiation is needed, the agents are trained and tested in order to receive a weight corresponding to their adequacy in the classification process. So, the roles of general and specialized classification agents are only responsible to classify the data and send this result to the


Fuzzy Logic on a Polygenic Multi-Agent System for Steganalysis of Digital Images 253

4. *short-line-density-5: counts how many lines with low contrast and size lesser or equal to 5* 

5. *short-line-density-2: counts how many lines with high contrast and size greater or equal to 5* 

The features above were used based in literature review, where we choose to operate in special domain instead of transform domain, by empirical experimentation. Though, we observed that the efficiency of the machine learning methods did not decrease by eliminating many of the features above, so they were excluded from the final features list.

To create the dataset, we utilized 300 images of landscapes, interiors, animals, buildings, people and food. According to the graphical complexity of each image, they were categorized as smooth, regular, complex or noisy. A random half of these images were kept unmodified, while the other half received hidden data corresponding up to 10% of the carry images size, what represents 20% of the maximum payload a carry image may cover (which is 50% of the total size of the carry image). The average size of the cover images is 800 x 600 pixels. And the stego objects here were created with the steganographic method

Then, this dataset was once again divided. A random 80% of all the images were separated

At runtime, when a specialized classifier agent is instantiated, it receives a random subset from the training set. This subset is selected from all the samples that correspond to one single of the four image categories used in this work (smooth, regular, complex, noisy). Three other specialized classifier agents are created to the other categories. The subset for each of these agents is 20% the size of the training set samples. Similarly, when a general

to compose the training set, and the 20% left composed the testing set.

1. *region-centroid-col: central column in a 3x3 pixel region;*  2. *region-centroid-row: central line in a 3x3 pixel region;*  3. *region-pixel-count: total of pixels in a 3x3 region = 9.* 

*passes through the region;* 

*passes through the region;*  6. *vedge-mean: vertical edge mean;* 

7. *vegde-sd: vertical edge standard-deviation;*  8. *hedge-mean: horizontal edge mean;* 

9. *hedge-sd: horizontal edge standard-deviation;*  10. *intensity-mean: (R + G + B)/3 in a region;*  11. *rawred-mean: red mean in a region;*  12. *rawblue-mean: blue mean in a region;*  13. *rawgreen-mean: green mena in a region;* 

14. *exred-mean: additional red mean: (2R - (G + B));*  15. *exblue-mean: additional blue mean: (2B - (G + R));*  16. *exgreen-mean: additional green mean: (2G - (R + B));*  17. *value-mean: non-linear 3D transformation mean;*  18. *saturation-mean: saturation mean in the 3D transform;* 

19. *hue-mean: hue mean in the 3D transform;* 

**3.3.3 Data samples** 

JPHide/JPSeek (Lathan 2006).

**3.3.4 Training and testing** 


where,

TAAR - this agent accuracy rate OAAR - other agent accuracy rate TC - class according to this agent OC - class according to the other agent high - accuracy rate is greater than or equal 0.8 medium - accuracy rate is greater than or equal 0.6 and lesser then 0.8 low - accuracy rate is lesser than 0.6 class - the new result of this agent accord - finish the negotiation process TAFN - this agent false negative rate OAFN - other agent false negative rate

#### **3.3 Description of steganalysis**

For the approach described here, we assume that suspicious data is already intercepted and submitted to this approach in order to verify if an object is a stego object or a regular file.

#### **3.3.1 Classifier method**

The classifier method used in this work is the Polygenic MAS Fuzzy Clustering. In a MAS architecture, coordination agents use fuzzy clustering inference to group the classification result of specific and general classification agents. Also, negotiation is performed between coordination agents in order to decide the better result.

The classifiers of two specific classification agents are divergent, both because each can be trained for a different type of image, and because each receive a different training subset of the data samples. This last reason also applies to describe the difference between two general classification agents.

The machine learning algorithm chosen for the internal classifier of each agent is a Decision Tree. But different machine learning algorithms would be applied to this architecture in order to search for a more robust classification.

#### **3.3.2 Features calibration**

The features that describe each instance or sample are: the four statistical momentums (mean, variance, asymmetry and kurtosis) for both the RGB and the HSBr matrixes, the image category (smooth, regular, complex, noisy), and the name of the class that sample describes (stego/non-stego).

This configuration of features was settled after some tryouts with bigger feature lists, which included:


17. IF TAAR=Low and OAAR=Low and TC!=OC and TAAR=OAAR and TAFN < OAFN

18. IF TAAR=Low and OAAR=Low and TC!=OC and TAAR=OAAR and TAFN >= OAFN

19. IF TAAR=Low and OAAR=Low and TC!=OC and TAAR<OAAR THEN TAAR=TAAR-

For the approach described here, we assume that suspicious data is already intercepted and submitted to this approach in order to verify if an object is a stego object or a regular file.

The classifier method used in this work is the Polygenic MAS Fuzzy Clustering. In a MAS architecture, coordination agents use fuzzy clustering inference to group the classification result of specific and general classification agents. Also, negotiation is performed between

The classifiers of two specific classification agents are divergent, both because each can be trained for a different type of image, and because each receive a different training subset of the data samples. This last reason also applies to describe the difference between two

The machine learning algorithm chosen for the internal classifier of each agent is a Decision Tree. But different machine learning algorithms would be applied to this architecture in

The features that describe each instance or sample are: the four statistical momentums (mean, variance, asymmetry and kurtosis) for both the RGB and the HSBr matrixes, the image category (smooth, regular, complex, noisy), and the name of the class that sample

This configuration of features was settled after some tryouts with bigger feature lists, which

THEN TAAR=TAAR+0.01

THEN TAAR=TAAR-0.01

TAAR - this agent accuracy rate OAAR - other agent accuracy rate TC - class according to this agent OC - class according to the other agent

low - accuracy rate is lesser than 0.6 class - the new result of this agent accord - finish the negotiation process TAFN - this agent false negative rate OAFN - other agent false negative rate

**3.3 Description of steganalysis** 

**3.3.1 Classifier method** 

general classification agents.

**3.3.2 Features calibration** 

describes (stego/non-stego).

included:

high - accuracy rate is greater than or equal 0.8

coordination agents in order to decide the better result.

order to search for a more robust classification.

medium - accuracy rate is greater than or equal 0.6 and lesser then 0.8

0.01

where,


The features above were used based in literature review, where we choose to operate in special domain instead of transform domain, by empirical experimentation. Though, we observed that the efficiency of the machine learning methods did not decrease by eliminating many of the features above, so they were excluded from the final features list.

#### **3.3.3 Data samples**

To create the dataset, we utilized 300 images of landscapes, interiors, animals, buildings, people and food. According to the graphical complexity of each image, they were categorized as smooth, regular, complex or noisy. A random half of these images were kept unmodified, while the other half received hidden data corresponding up to 10% of the carry images size, what represents 20% of the maximum payload a carry image may cover (which is 50% of the total size of the carry image). The average size of the cover images is 800 x 600 pixels. And the stego objects here were created with the steganographic method JPHide/JPSeek (Lathan 2006).

Then, this dataset was once again divided. A random 80% of all the images were separated to compose the training set, and the 20% left composed the testing set.

#### **3.3.4 Training and testing**

At runtime, when a specialized classifier agent is instantiated, it receives a random subset from the training set. This subset is selected from all the samples that correspond to one single of the four image categories used in this work (smooth, regular, complex, noisy). Three other specialized classifier agents are created to the other categories. The subset for each of these agents is 20% the size of the training set samples. Similarly, when a general

Fuzzy Logic on a Polygenic Multi-Agent System for Steganalysis of Digital Images 255

As future work we intend to to improve this paradigm once the use of MAS can be extended to other learning techniques besides fuzzy logic. A comparison between several techniques will be performed and a possible solution combining two or several of them will also be tried in order to achieve even better performance. For example, we believe that the use of decision trees combined to our fuzzy approach depicgted here can be used hopefully to get better results. So a possible future direction for our work is to test this approach with other techniques and also to use other medias as video, text, and audio, not being addressed in this work. Finallyanother possibility is to develop a more complete system including

Acampora 2011 Acampora, G.; Cadenas, J.M.; Loia, V.; Ballester, E.M.;A Multi-Agent

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on the Application of Expert Systems, v. 28 (4), pp 339–352, September 2011. Artz 2001 ARTZ, Donovan. Digital Steganography: Hiding Data within Data. IEEE Internet

Bakhshandeh 2009 Bakhshandeh, Soodeh; Jamjah, Javad Ravan; Azami, Bahram Zahir: Blind

Carpenter 1992 Carpenter, G.A.; Grossberg, S.; Markuzon, N.; Reynolds, J.H.; Rosen, D.B.:

Chandramouli 2004 Chandramouli, R.; Subbalakshmi, K.P: Current Trends in Steganalysis:

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Goodarzi 2011 Goodarzi, Mohammad; Radmand, Ashkan; Nazemi, Eslam: An Optimized

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techniques for extracting the hidden information.

Vol.20, Iss.1, pp.013016, 2011, ISSN: 10179909

Universidade de São Paulo, 2003.

Computings, v.5, n.3, mai.-jun. 2001.

SPRINGER-VERLAG, BERLIN, 2009.

2004. (invited paper)

Contents , 2004.

no.5, pp.698-713, Sep 1992.doi: 10.1109/72.159059

**6. References** 

960, Sept. 2011.

classifier agent is instantiated, it receives a random subset from the training dataset that corresponds to 40% of the total size of the training dataset.

Then, these agents train theirs classifiers and test their performances with 20% of the training dataset (the testing dataset is for testing the approach as a whole). The resulting accuracy rate will be informed to the coordinator agent, generating a weight to that agent decision, and influencing the fuzzy inference mechanism.

The size of these datasets was limited to this small percent in order to produce different agents, with different performances, that will be combined by the polygenic MAS fuzzy clustering approach.

#### **4. Experiments and results**

We have developed experiments and tests following the planning experimentation setup discussed by Cobb (1997). Basically, this methodology is resumed in *what measures to take, under what conditions and which material to process in the testes*. The answers are the measures given by the MAS about the classification: correctness rate, false positive and false negative. A set of training data is presented to the system, which is randomly distributed into other data sets for training the classifier agents. The test data is then distributed between the coordinator agents in order for these to coordinate classification activities of the general classifier agents and the specific classifier agents. Finally, the cited rates are obtained and analyzed.

As result of these experiments, the system presents a rate for correct detection of 89,37%, with false positive of 10,63% and false negative of 10,54%.

For JPHide and Seek, Liu (2008) present accuracy rate of 0.8% with OAASVM, and 56% with Adaboost. It is important to say that their dataset is different from the one used here, although we may say that our experiment presented a considerate accuracy rate.

Bakhshandeh (2009) presented accuracy rates from 68,75% to 94,67%, but none of the steganography methods used in their experiment was the same used in our work or in Lius´.

Also, we trained a Decision tree without the Polygenic MAS Fuzzy Clustering Steganalysis, using the entire dataset. And the results were an accuracy rate of 72,45%, false positives of 27,23% and false negatives of 26,92%.

These results show that the Polygenic MAS Fuzzy Clustering Steganalysis approach increased the performance of a machine learning steganalysis.

#### **5. Conclusion**

We have proposed a useful technique to detect images that possibly carry encrypted data on its contents. We use a methodology based on polygenic bees (a model based on community) where several agents interact between them. Our model combines this multi-agent system system with fuzzy logic in order to decide whether a digital media object has hidden information, coming up with a decision at the end of processed interactions. In comparison to the rates of correctness of other techniques found in the literature (between 70% to 90%) our rates of about 89% indicates that the proposed approach based on fuzzy logic is a good choice in this direction, being efficient in this task, experimentally comproved.

As future work we intend to to improve this paradigm once the use of MAS can be extended to other learning techniques besides fuzzy logic. A comparison between several techniques will be performed and a possible solution combining two or several of them will also be tried in order to achieve even better performance. For example, we believe that the use of decision trees combined to our fuzzy approach depicgted here can be used hopefully to get better results. So a possible future direction for our work is to test this approach with other techniques and also to use other medias as video, text, and audio, not being addressed in this work. Finallyanother possibility is to develop a more complete system including techniques for extracting the hidden information.

#### **6. References**

254 Fuzzy Logic – Emerging Technologies and Applications

classifier agent is instantiated, it receives a random subset from the training dataset that

Then, these agents train theirs classifiers and test their performances with 20% of the training dataset (the testing dataset is for testing the approach as a whole). The resulting accuracy rate will be informed to the coordinator agent, generating a weight to that agent

The size of these datasets was limited to this small percent in order to produce different agents, with different performances, that will be combined by the polygenic MAS fuzzy

We have developed experiments and tests following the planning experimentation setup discussed by Cobb (1997). Basically, this methodology is resumed in *what measures to take, under what conditions and which material to process in the testes*. The answers are the measures given by the MAS about the classification: correctness rate, false positive and false negative. A set of training data is presented to the system, which is randomly distributed into other data sets for training the classifier agents. The test data is then distributed between the coordinator agents in order for these to coordinate classification activities of the general classifier agents and the specific classifier agents. Finally, the cited rates are obtained and

As result of these experiments, the system presents a rate for correct detection of 89,37%,

For JPHide and Seek, Liu (2008) present accuracy rate of 0.8% with OAASVM, and 56% with Adaboost. It is important to say that their dataset is different from the one used here,

Bakhshandeh (2009) presented accuracy rates from 68,75% to 94,67%, but none of the steganography methods used in their experiment was the same used in our work or in Lius´. Also, we trained a Decision tree without the Polygenic MAS Fuzzy Clustering Steganalysis, using the entire dataset. And the results were an accuracy rate of 72,45%, false positives of

These results show that the Polygenic MAS Fuzzy Clustering Steganalysis approach

We have proposed a useful technique to detect images that possibly carry encrypted data on its contents. We use a methodology based on polygenic bees (a model based on community) where several agents interact between them. Our model combines this multi-agent system system with fuzzy logic in order to decide whether a digital media object has hidden information, coming up with a decision at the end of processed interactions. In comparison to the rates of correctness of other techniques found in the literature (between 70% to 90%) our rates of about 89% indicates that the proposed approach based on fuzzy logic is a good

choice in this direction, being efficient in this task, experimentally comproved.

although we may say that our experiment presented a considerate accuracy rate.

corresponds to 40% of the total size of the training dataset.

decision, and influencing the fuzzy inference mechanism.

with false positive of 10,63% and false negative of 10,54%.

increased the performance of a machine learning steganalysis.

clustering approach.

analyzed.

**4. Experiments and results** 

27,23% and false negatives of 26,92%.

**5. Conclusion** 


**Part 3** 

**Business, Environment and Energy** 

Studies in Computational Intelligence, 2011, V. 325, pp 377-388, DOI: 10.1007/978- 3-642-16098-1\_23


**Part 3** 

**Business, Environment and Energy** 

256 Fuzzy Logic – Emerging Technologies and Applications

Hagras 2010 Hagras, H.; Ramadan, R.; Nawito, M.; Gabr, H.; Zaher, M.; Fahmy, H.: A fuzzy

Jang 1993 Jang, JSR: Anfis - Adaptive-Network-Based Fuzzy Inference System. in IEEE

Katzenbeisser 2000 Katzenbeisser, Stefan; Petitcolas, Fabien A. P. Information Hiding

Kofler 2003 Kofler, R. Krimmer, R. Prosser, A.: Electronic Voting: Algorithmic and

Lin 2002 Lin, Chun-Fu; Wang, Sheng-De.: Fuzzy support vector machines in IEEE Transactions on Neural Networks, vol.13, no.2, pp.464-471, Mar 2002. Liu 2008 liu, qingzhong; sung, Andrew H.: Detect Information-Hiding Type and Length in

Liu 2008b Liu, Q., Sung, A.H., Chen, Z., Xu, J.: Feature mining and pattern classification for

López-Ortega 2011 López-Ortega, Omar; Rosales, Marco-Antonio: An agent-oriented

Macq 1995 Macq, B. M.; Quisquater , J-J. Cryptology for digital TV broadcasting.

Moskowitz 2002 MOSKOWITZ, Ira S.; CHANG, Liwu; NEWMAN, Richard E. Capacity is

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Sanches 2004 Sanches, M. K.; Geromini, M. R.; Aprendizado de Máquina: Relatório Técnico.

Wooldridge 2001 Woolridge, Michael J., Introduction to Multiagent Systems, John Wiley &

Zadeh 1965 Zadeh, L. A.: Fuzzy sets. Information and Control, 8(3), pp. 338-353, 1965.

Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, 2004.

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based hierarchical coordination and control system for a robotic agent team in the robot Hockey competition, in IEEE International Conference on Fuzzy Systems

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**13** 

*Poland* 

Tomasz Korol

*Gdansk University of Technology,* 

**Fuzzy Logic in Financial Management** 

Fuzzy logic has been widely used in machinery, robotics, and industrial engineering. This chapter introduces the use of fuzzy logic for the needs of financial management. The process of globalization has led to the emergence of a complex network of relationships in the business environment. In a free market economy, this means increased complexity and uncertainty of factors affecting the financial standing of entities. Nowadays many phenomena in finance and economics are fuzzy, but are treated as if they were crisp. In this chapter two such financial research problems are analyzed. The first concerns the issue of consumer credit scoring, while the second the forecasting of the financial situation of firms in short and medium periods (one year and two years forecasts). Predicting both business and consumer bankruptcy, is imprecise and ambiguous. The failure process is affected by many internal and external factors that cannot be precisely and unambiguously defined. Also, the mere allegation that a company or an individual consumer is at risk of bankruptcy must be considered imprecise, and in fact rarely in economic reality are there firms/persons that can be considered as 100% bankrupt. It is difficult to accurately determine the degree of bankruptcy threat using traditional statistical methods such as multivariate discriminant analysis. When the value of the discriminant function is less than the threshold value, we find that a company is at risk of bankruptcy. With the use of fuzzy logic vague and ambiguous concepts can be defined, such as "high risk of bankruptcy" or "low risk of bankruptcy". The presented models are the result of the chapter author's ten years of experience on this issue. They can be used not only for forecasting the level of risk of bankruptcy but also for determining the degree of positive financial standing of the analyzed entity (a company or consumer) – for example, such as "outstanding solvency" or "average solvency" etc. The global financial crisis that began in mid-2008 caused the number of companies in danger of bankruptcy to significantly increase around the world. Furthermore, the highly globalized environment has caused the economies of countries to deteriorate too (for example: such countries as Greece or Iceland risking bankruptcy; the decrease of the USA's credit rating from AAA to AA+ by rating agencies for the first time in history), which directly and indirectly influences the financial situation of both companies and consumers. Therefore, analysts are no longer faced with the dilemma of whether to predict the financial standing of entities (enterprises, consumers, or even countries), but

what forecasting method to use in order to minimize forecast errors.

This chapter consists of three sections. In the first the author introduces his financial forecasting methodology and describes the concept of using fuzzy logic in finance. Section 2

**1. Introduction** 

### **Fuzzy Logic in Financial Management**

### Tomasz Korol

*Gdansk University of Technology, Poland* 

#### **1. Introduction**

Fuzzy logic has been widely used in machinery, robotics, and industrial engineering. This chapter introduces the use of fuzzy logic for the needs of financial management. The process of globalization has led to the emergence of a complex network of relationships in the business environment. In a free market economy, this means increased complexity and uncertainty of factors affecting the financial standing of entities. Nowadays many phenomena in finance and economics are fuzzy, but are treated as if they were crisp. In this chapter two such financial research problems are analyzed. The first concerns the issue of consumer credit scoring, while the second the forecasting of the financial situation of firms in short and medium periods (one year and two years forecasts). Predicting both business and consumer bankruptcy, is imprecise and ambiguous. The failure process is affected by many internal and external factors that cannot be precisely and unambiguously defined. Also, the mere allegation that a company or an individual consumer is at risk of bankruptcy must be considered imprecise, and in fact rarely in economic reality are there firms/persons that can be considered as 100% bankrupt. It is difficult to accurately determine the degree of bankruptcy threat using traditional statistical methods such as multivariate discriminant analysis. When the value of the discriminant function is less than the threshold value, we find that a company is at risk of bankruptcy. With the use of fuzzy logic vague and ambiguous concepts can be defined, such as "high risk of bankruptcy" or "low risk of bankruptcy". The presented models are the result of the chapter author's ten years of experience on this issue. They can be used not only for forecasting the level of risk of bankruptcy but also for determining the degree of positive financial standing of the analyzed entity (a company or consumer) – for example, such as "outstanding solvency" or "average solvency" etc. The global financial crisis that began in mid-2008 caused the number of companies in danger of bankruptcy to significantly increase around the world. Furthermore, the highly globalized environment has caused the economies of countries to deteriorate too (for example: such countries as Greece or Iceland risking bankruptcy; the decrease of the USA's credit rating from AAA to AA+ by rating agencies for the first time in history), which directly and indirectly influences the financial situation of both companies and consumers. Therefore, analysts are no longer faced with the dilemma of whether to predict the financial standing of entities (enterprises, consumers, or even countries), but what forecasting method to use in order to minimize forecast errors.

This chapter consists of three sections. In the first the author introduces his financial forecasting methodology and describes the concept of using fuzzy logic in finance. Section 2

Fuzzy Logic in Financial Management 261

observations for each individual object (solvent and insolvent companies/clients) must

object classifications must be clearly defined – belonging to one group excludes its

be complete – i.e. should have values for all indicators of all entities,

Table 1. Classification of Forecasting Models (the source: based on own studies)

In contrast to the statistical models, methods of soft computing techniques effectively cope with imprecisely defined problems, incomplete data, imprecision, and uncertainty. The issue of consumer and business bankruptcy prediction has all of the above characteristics. In addition, soft computing models are suitable for use in dynamic systems designed to fit certain internal parameters to changing environmental conditions (so-called learning systems). The difference between statistical models and soft computing models is based on aspects such as the precision, reliability, and accuracy of variables used. These elements are the basis of statistical models, while the starting point, e.g. for the fuzzy logic model, is the thesis that precision and certainty carry a cost, and calculating, reasoning, and decision making should exploit tolerance for imprecision and uncertainty wherever possible. Soft computing techniques, in contrast to statistical models, thus tolerate inaccurate data, uncertainty, and approximation. The essence of models based on computational intelligence is the processing and interpretation of data in a variety of capacities. They are able to formulate rules of inference and generalized knowledge about situations where they are expected to predict or classify the object into one of the previously observed categories.

The theoretical models are mainly focused on the use of qualitative information in predicting the bankruptcy of entities. In contrast to the statistical and soft computing methods that rely on the symptoms of going bankrupt, theoretical models focus on finding the causes of the collapse. Theoretical models typically use different statistical techniques for drawing conclusions and quantitative proof of the theoretical argument. Thus, for example in the hazard model, an entity can be seen from the perspective of the player – gambler who

belonging to a second group.

is devoted to the author's research on the use of fuzzy logic in consumer credit scoring. Models developed by the author are based on demographic and financial variables of customers of a Central European bank. In the last section, the author presents business bankruptcy prediction models programmed by him. These models are based on financial variables of companies quoted on a stock exchange in Central Europe.

The information contained in this chapter may be used in practice in several aspects:


#### **2. Methodology of financial forecasting**

#### **2.1 Classification of financial forecasting models**

In literature, forecasting models are categorized into three main groups: statistical models, theoretical models, and models using soft computing techniques, which are part of a separate field of science defined as *Computational Intelligence* (a term understood as solving various problems with the help of artificial intelligence). According to literature, 64% of case studies used statistical models, 25% soft computing techniques, and 11% other types of models (Aziz & Dar, 2006).

In statistical models, selected financial ratios that have diagnostic value are estimated and used. The selection of each ratio is based on empirical studies of ex-post groups of entities, consisting of enterprises/consumers with good financial condition and those at risk. Furthermore, the set of indicators is reduced by excluding variables of similar information content, e.g. ratios that are correlated with each other. After defining a set of diagnostic variables, the model's parameters are estimated. Each variable selected receives discriminatory weight. The bankruptcy prediction model is created by a gradual "compaction" of the set of individual ratios, to obtain a single index called a synthetic indicator. "Compaction" is carried out using appropriate statistical and econometrical methods. Using such a model for assessing the risk of bankruptcy is the substitution of the actual value of financial ratios and the calculation of the synthetic indicator of risk. This synthetic index characterizes the financial situation of the audited company/client.

The use of statistical models requires that the variables used in the model meet the following assumptions:


is devoted to the author's research on the use of fuzzy logic in consumer credit scoring. Models developed by the author are based on demographic and financial variables of customers of a Central European bank. In the last section, the author presents business bankruptcy prediction models programmed by him. These models are based on financial

in the context of early warning of the deteriorating financial situation of an audited

In literature, forecasting models are categorized into three main groups: statistical models, theoretical models, and models using soft computing techniques, which are part of a separate field of science defined as *Computational Intelligence* (a term understood as solving various problems with the help of artificial intelligence). According to literature, 64% of case studies used statistical models, 25% soft computing techniques, and 11% other types of

In statistical models, selected financial ratios that have diagnostic value are estimated and used. The selection of each ratio is based on empirical studies of ex-post groups of entities, consisting of enterprises/consumers with good financial condition and those at risk. Furthermore, the set of indicators is reduced by excluding variables of similar information content, e.g. ratios that are correlated with each other. After defining a set of diagnostic variables, the model's parameters are estimated. Each variable selected receives discriminatory weight. The bankruptcy prediction model is created by a gradual "compaction" of the set of individual ratios, to obtain a single index called a synthetic indicator. "Compaction" is carried out using appropriate statistical and econometrical methods. Using such a model for assessing the risk of bankruptcy is the substitution of the actual value of financial ratios and the calculation of the synthetic indicator of risk. This

synthetic index characterizes the financial situation of the audited company/client.

The use of statistical models requires that the variables used in the model meet the following

indicators must have a high discriminative ability of separating solvent entities from

 in the context of the implementation of financial and economic plans in a company, from the perspective of risk assessment, the purchase of shares by individual and

in the context of credit scoring the credit applications of consumers by banks,

from the viewpoint of assessing the consumer bankruptcy threat.

The information contained in this chapter may be used in practice in several aspects:

 from the viewpoint of assessing the solvency of partners and customers, from the perspective of credit risk assessment by financial institutions,

variables of companies quoted on a stock exchange in Central Europe.

institutional investors on stock exchanges,

**2. Methodology of financial forecasting** 

**2.1 Classification of financial forecasting models** 

indicators should have normal distributions,

indicators must be independent,

insolvent ones,

company,

models (Aziz & Dar, 2006).

assumptions:


Table 1. Classification of Forecasting Models (the source: based on own studies)

In contrast to the statistical models, methods of soft computing techniques effectively cope with imprecisely defined problems, incomplete data, imprecision, and uncertainty. The issue of consumer and business bankruptcy prediction has all of the above characteristics. In addition, soft computing models are suitable for use in dynamic systems designed to fit certain internal parameters to changing environmental conditions (so-called learning systems). The difference between statistical models and soft computing models is based on aspects such as the precision, reliability, and accuracy of variables used. These elements are the basis of statistical models, while the starting point, e.g. for the fuzzy logic model, is the thesis that precision and certainty carry a cost, and calculating, reasoning, and decision making should exploit tolerance for imprecision and uncertainty wherever possible. Soft computing techniques, in contrast to statistical models, thus tolerate inaccurate data, uncertainty, and approximation. The essence of models based on computational intelligence is the processing and interpretation of data in a variety of capacities. They are able to formulate rules of inference and generalized knowledge about situations where they are expected to predict or classify the object into one of the previously observed categories.

The theoretical models are mainly focused on the use of qualitative information in predicting the bankruptcy of entities. In contrast to the statistical and soft computing methods that rely on the symptoms of going bankrupt, theoretical models focus on finding the causes of the collapse. Theoretical models typically use different statistical techniques for drawing conclusions and quantitative proof of the theoretical argument. Thus, for example in the hazard model, an entity can be seen from the perspective of the player – gambler who

Fuzzy Logic in Financial Management 263

where μA : X [0,1] is a function for each element of X that determines the extent to which

Classical set theory assumes that any element (company) fully belongs or completely does not belong to a given set (bankrupt or non-bankrupt set of companies). In turn, in the fuzzy set theory an element (company) may partially belong to a certain set, and this membership may be expressed by means of a real number in the interval [0,1]. Thus, the membership

f(x),x X (x) 0,x X

where: μA(x) –function defining membership of element x to set A, which is a subset of U; f(x) - function receiving values from the interval [0,1]. The values of this function are called

A membership function assigns the degree of membership of each element x X to a fuzzy

Membership functions are usually presented in graphical form. A trapezoidal function μ<sup>A</sup> (x) is often used (see Figure 1). The graph shows information from literature about the accepted values of the cash liquidity ratio. The correct values for this ratio are values in the interval [0.2, 0.5], and incorrect values are in the range of (0; 0.2)(0.5,). When this ratio is lower than 0.2 it is considered that the company has a cash liquidity shortage; in turn, when this amount is higher than 0.5 it is said that the company has excess liquidity, which is also rated as a negative phenomenon (in the case of excess liquidity such companies have too

Fig. 1. An Example of the Trapezoidal Membership Function for the Cash Liquidity Ratio

Value of the cash liquidity

i

INCORRECT (TOO HIGH)

it belongs to set A. This function is called a membership function of fuzzy set A.

<sup>A</sup> x U

μA (x) = 1 means full membership of element x to the fuzzy set A,

0< μA (x) <1 means partial membership of an element x to the fuzzy set A.

μA (x) = 0 means that no element x belongs to fuzzy set A,

much cash, which is rated as inefficient company management).

0.2 0.5

 

function μA(x) : U [0,1] is defined as follows:

set A, where we can distinguish three situations:

the degrees of membership.

Degree of membership

(TOO LOW)

1

0

*μ* (value of the cash liquidity ratio)

INCORRECT CORRECT

plays burdened with a certain probability of loss. The player (company/consumer) continues to function until the moment when its net worth reaches zero (bankruptcy). Another example of the theoretical model is the KMV model, which is based on the use of option pricing theory for the valuation of risky loans and bonds. In the KMV model an entity's net assets are essential. This model assumes that at any time the value of assets can be modelled as a call option whose underlying is the market value of company assets and the exercise price – the value of the entity's liabilities at the time of their maturity. Using the KMV model the probability of a company's value falling below the value of its liabilities (making the firm insolvent) can be determined, .

Literature studies show that the financial situation predictions are dominated by discriminant analysis models, which make up 30.3 percent of all models created among all methods – statistical, soft computing, and theoretical (Aziz & Dar, 2006). Undoubtedly the most popular model for forecasting bankruptcy risk is the statistical model developed by American Professor – E. Altman in 1968. As a pioneer in the use of multivariate discriminant analysis to predict the bankruptcy of companies, he developed a model consisting of a single function with five financial ratios (Altman, 1993):

$$Z = 1.2 \, ^\ast X\_1 + 1.4 \, ^\ast X\_2 + 3.3 \, ^\ast X\_3 + 0.6 \, ^\ast X\_4 + 0.999 \, ^\ast X\_5 \tag{1}$$

where:

X1 = working capital / total assets X2 = retained earnings / total assets X3 = earnings before taxes / total assets X4 = market value of equity / total long term and short term liabilities X5 = sales / total assets

Altman proposed the use of three decision areas depending on the value of the Z score:


Predicting the bankruptcy of companies is imprecise and ambiguous. The process of business failure is affected by many internal and external factors that cannot be precisely and unambiguously defined. Also, the mere allegation that a company is at risk of bankruptcy must be considered imprecise, and in fact rarely in economic reality are there companies that can be considered as 100% bankrupt. It is difficult to accurately determine the degree of bankruptcy threat using traditional statistical methods such as multivariate discriminant analysis. When the value of the discriminant function is less than the threshold value, we find that a company is at risk of bankruptcy. With the use of fuzzy logic vague and ambiguous concepts can be defined, such as "high risk of bankruptcy" or "low risk of bankruptcy". The concept of fuzzy sets was introduced by Zadeh in 1965 (Zadeh, 1965). The fuzzy set "A" in a non-empty space X (AX) can be defined as:

$$A = \{ (\mathbf{x}, \mu\_A \left( \mathbf{x} \right)) \mid \mathbf{x} \in \mathcal{X} \}\tag{2}$$

plays burdened with a certain probability of loss. The player (company/consumer) continues to function until the moment when its net worth reaches zero (bankruptcy). Another example of the theoretical model is the KMV model, which is based on the use of option pricing theory for the valuation of risky loans and bonds. In the KMV model an entity's net assets are essential. This model assumes that at any time the value of assets can be modelled as a call option whose underlying is the market value of company assets and the exercise price – the value of the entity's liabilities at the time of their maturity. Using the KMV model the probability of a company's value falling below the value of its liabilities

Literature studies show that the financial situation predictions are dominated by discriminant analysis models, which make up 30.3 percent of all models created among all methods – statistical, soft computing, and theoretical (Aziz & Dar, 2006). Undoubtedly the most popular model for forecasting bankruptcy risk is the statistical model developed by American Professor – E. Altman in 1968. As a pioneer in the use of multivariate discriminant analysis to predict the bankruptcy of companies, he developed a model consisting of a

Z = 1.2 \* X1 + 1.4 \* X2 + 3.3 \* X3 + 0.6 \* X4 + 0.999 \* X5 (1)

A = {(x, μA (x))| x X } (2)

(making the firm insolvent) can be determined, .

single function with five financial ratios (Altman, 1993):

X4 = market value of equity / total long term and short term liabilities

if Z < 1.81 then it is a signal of a high probability of bankruptcy,

if Z > 2.99 then there is low probability of bankruptcy.

fuzzy set "A" in a non-empty space X (AX) can be defined as:

Altman proposed the use of three decision areas depending on the value of the Z score:

if 1.81 < Z < 2.99 then the risk of financial failure of the company is not possible to

Predicting the bankruptcy of companies is imprecise and ambiguous. The process of business failure is affected by many internal and external factors that cannot be precisely and unambiguously defined. Also, the mere allegation that a company is at risk of bankruptcy must be considered imprecise, and in fact rarely in economic reality are there companies that can be considered as 100% bankrupt. It is difficult to accurately determine the degree of bankruptcy threat using traditional statistical methods such as multivariate discriminant analysis. When the value of the discriminant function is less than the threshold value, we find that a company is at risk of bankruptcy. With the use of fuzzy logic vague and ambiguous concepts can be defined, such as "high risk of bankruptcy" or "low risk of bankruptcy". The concept of fuzzy sets was introduced by Zadeh in 1965 (Zadeh, 1965). The

X1 = working capital / total assets X2 = retained earnings / total assets X3 = earnings before taxes / total assets

define (it is a so-called "gray area"),

X5 = sales / total assets

where:

where μA : X [0,1] is a function for each element of X that determines the extent to which it belongs to set A. This function is called a membership function of fuzzy set A.

Classical set theory assumes that any element (company) fully belongs or completely does not belong to a given set (bankrupt or non-bankrupt set of companies). In turn, in the fuzzy set theory an element (company) may partially belong to a certain set, and this membership may be expressed by means of a real number in the interval [0,1]. Thus, the membership function μA(x) : U [0,1] is defined as follows:

$$\biguplus\_{\mathbf{x}\in\mathcal{U}}\mu\_{\mathcal{A}}(\mathbf{x}) = \begin{cases} \mathbf{f}(\mathbf{x}), \mathbf{x} \in \mathcal{X} \\ \mathbf{0}, \mathbf{x} \notin \mathcal{X} \end{cases}$$

where: μA(x) –function defining membership of element x to set A, which is a subset of U; f(x) - function receiving values from the interval [0,1]. The values of this function are called the degrees of membership.

A membership function assigns the degree of membership of each element x X to a fuzzy set A, where we can distinguish three situations:


Membership functions are usually presented in graphical form. A trapezoidal function μ<sup>A</sup> (x) is often used (see Figure 1). The graph shows information from literature about the accepted values of the cash liquidity ratio. The correct values for this ratio are values in the interval [0.2, 0.5], and incorrect values are in the range of (0; 0.2)(0.5,). When this ratio is lower than 0.2 it is considered that the company has a cash liquidity shortage; in turn, when this amount is higher than 0.5 it is said that the company has excess liquidity, which is also rated as a negative phenomenon (in the case of excess liquidity such companies have too much cash, which is rated as inefficient company management).

Fig. 1. An Example of the Trapezoidal Membership Function for the Cash Liquidity Ratio

i

Fuzzy Logic in Financial Management 265

OLD PEOPLE

Fig. 2. An Example of the Trapezoidal Membership Function for the Age of Consumers.

Despite the high popularity of traditional bankruptcy prediction models, they are not free of defects and limitations, which rarely receive substantive discussion in literature. The first limitation has already been discussed – the crisp separation between "good" and "bad" values, conditions or situations. Such models use classical logic with no possible partial

The age of the consumer

The second issue in assessing the effectiveness of these models is the method of developing a learning dataset (based on which the model shall be estimated) and a testing dataset that consists of entities that did not make it into the learning sample. Elements of the testing sample are unknown to the model. It enables evaluating the effectiveness of the model in conditions similar to those in business practice. In literature, the vast majority of scientists (e.g.: Ooghe & Balcaen, 2006; or Kumar & Ravi, 2007) suggest that the learning dataset was a balanced sample (consisting 50% of entities at risk of bankruptcy, and 50% of entities in good financial condition). This will enable the model to learn to distinguish "good" and "bad" entities. Note, however, that in a market economy the number of firms/consumers at risk of bankruptcy is much smaller than the number of "healthy" entities. Evaluation of the effectiveness of models that use a balanced testing dataset become highly questionable. After all, these models are developed for use in business practice, where the proportion of bankrupts to non-bankrupts is many times smaller. The author of this chapter proved in his previous research that fuzzy logic models are superior over traditional bankruptcy prediction models (both statistical and soft computing techniques) in forecasting risk of

bankruptcy of companies in the case of an unbalanced testing dataset (Korol, 2011).

Another controversial aspect on the effectiveness of the most popular analysis methods – multivariate discriminant, logit, and probit , is the possibility of manipulation of the threshold in order to maximize the classification results of these models. This allegation was raised by M. Nwogugu. According to him, the statistical methods do not guarantee reliable results because of the ease at which the threshold which separates "good" and "bad" entities can be manually set (Nwogugu, 2007). Such manipulation, of course, does not increase the effectiveness of the model in business practice after its implementation in a bank, but only in

**2.2 Drawbacks and limitations of traditional forecasting models** 

25 50

30 45

MIDDLE AGE PEOPLE

belonging to a defined group of criteria.

Degree of membership *μ* (the age of consumer)

> YOUNG PEOPLE

1

0

theoretical tests in literature.

In this case, using the classical set theory to evaluate this financial ratio, there is a sharp boundary between the two sets of ratio values 0.2 and 0.5. If one company recorded a cash liquidity ratio of 0.19, it would be classified as an incorrect value - negative, while if a second company recorded this ratio at the level of 0.2, it would be regarded as a correct value – positive assessment of bankruptcy risk, even though the financial ratio of the two firms differ only by 0.01. The interpretation of the values of individual ratios (e.g. liquidity) is further complicated by the fact that different literature sources give different reference limit values for individual financial ratios.

Application of a fuzzy set changes the assessment of the problem. A cash liquidity ratio with a value of 0.19 is considered as partly correct and partly invalid. The degree of membership to both sets depends on the shape of the membership function.

With such defined subsets, the boundary between the values considered to be positive or negative, is fuzzyficated – a certain ratio value is "partially good" and "partially bad." There is no such possibility in the case of classical logic, i.e. bivalent, in which the value of the ratio is "good" or "bad". Therefore, the use of classical logic in assessing the financial situation of companies affect negatively on the effectiveness of posed forecasts. This occurs especially in ratios which values are close to the threshold of subsets, where an excess of the critical value determines the final evaluation of the ratio (as entirely positive or negative), which is not true, because both values reflect almost the same situation in the enterprise.

The above example concerns the prediction of bankruptcy of companies. But the example for the usefulness of fuzzy logic in assessing the creditworthiness of consumers can also be given. In consumer credit scoring different demographical and financial variables of consumers are taken into account. Bank analysts set individual criteria to each of them in order to evaluate the credit risk of the applicant (setting certain points to each variable). One of the most popular factors is the age of the consumer.

It is generally accepted that the middle aged consumers group is less risky (young people tend to have smaller and less stable income than middle aged men, and old consumers bear higher risk because of their life expectancy). The issue is to set proper age limits into each category. Using the most common classical logic it can be set that middle aged consumers are those in the range of 30-45 years old. In such case a credit applicant who is 29 years old is evaluated on a scoring card worse than the consumer who is only 1 year older. The drawbacks of using classical logic are not only for the bank's clients who may not receive the credit but also for the bank itself that looses the potential profits from refused credit, which could have been given without much larger risk than in case of middle age people. Application of fuzzy logic can improve the efficiency in forecasting the probability of ontime repayment of granted credits. Figure 2 shows that classical logic uses crisp classification of the age of customers – group of young people in age ranges of (0; 30), group of old people in age ranges of (45 and more). With the help of fuzzy logic a bank can set that consumers with an age between 25 and 30 are partially young and middle age ones, and with an age between 45 and 50 are partially middle age and old ones. In the described example, the credit applicants who are 29 years old will be scored very similarly to those who are 30 years old, which would not be possible using credit scoring applications that are based on classical logic.

In this case, using the classical set theory to evaluate this financial ratio, there is a sharp boundary between the two sets of ratio values 0.2 and 0.5. If one company recorded a cash liquidity ratio of 0.19, it would be classified as an incorrect value - negative, while if a second company recorded this ratio at the level of 0.2, it would be regarded as a correct value – positive assessment of bankruptcy risk, even though the financial ratio of the two firms differ only by 0.01. The interpretation of the values of individual ratios (e.g. liquidity) is further complicated by the fact that different literature sources give different reference

Application of a fuzzy set changes the assessment of the problem. A cash liquidity ratio with a value of 0.19 is considered as partly correct and partly invalid. The degree of membership

With such defined subsets, the boundary between the values considered to be positive or negative, is fuzzyficated – a certain ratio value is "partially good" and "partially bad." There is no such possibility in the case of classical logic, i.e. bivalent, in which the value of the ratio is "good" or "bad". Therefore, the use of classical logic in assessing the financial situation of companies affect negatively on the effectiveness of posed forecasts. This occurs especially in ratios which values are close to the threshold of subsets, where an excess of the critical value determines the final evaluation of the ratio (as entirely positive or negative), which is not

The above example concerns the prediction of bankruptcy of companies. But the example for the usefulness of fuzzy logic in assessing the creditworthiness of consumers can also be given. In consumer credit scoring different demographical and financial variables of consumers are taken into account. Bank analysts set individual criteria to each of them in order to evaluate the credit risk of the applicant (setting certain points to each variable). One

It is generally accepted that the middle aged consumers group is less risky (young people tend to have smaller and less stable income than middle aged men, and old consumers bear higher risk because of their life expectancy). The issue is to set proper age limits into each category. Using the most common classical logic it can be set that middle aged consumers are those in the range of 30-45 years old. In such case a credit applicant who is 29 years old is evaluated on a scoring card worse than the consumer who is only 1 year older. The drawbacks of using classical logic are not only for the bank's clients who may not receive the credit but also for the bank itself that looses the potential profits from refused credit, which could have been given without much larger risk than in case of middle age people. Application of fuzzy logic can improve the efficiency in forecasting the probability of ontime repayment of granted credits. Figure 2 shows that classical logic uses crisp classification of the age of customers – group of young people in age ranges of (0; 30), group of old people in age ranges of (45 and more). With the help of fuzzy logic a bank can set that consumers with an age between 25 and 30 are partially young and middle age ones, and with an age between 45 and 50 are partially middle age and old ones. In the described example, the credit applicants who are 29 years old will be scored very similarly to those who are 30 years old, which would not be possible using credit scoring applications that are

limit values for individual financial ratios.

to both sets depends on the shape of the membership function.

of the most popular factors is the age of the consumer.

based on classical logic.

true, because both values reflect almost the same situation in the enterprise.

Fig. 2. An Example of the Trapezoidal Membership Function for the Age of Consumers.

#### **2.2 Drawbacks and limitations of traditional forecasting models**

Despite the high popularity of traditional bankruptcy prediction models, they are not free of defects and limitations, which rarely receive substantive discussion in literature. The first limitation has already been discussed – the crisp separation between "good" and "bad" values, conditions or situations. Such models use classical logic with no possible partial belonging to a defined group of criteria.

The second issue in assessing the effectiveness of these models is the method of developing a learning dataset (based on which the model shall be estimated) and a testing dataset that consists of entities that did not make it into the learning sample. Elements of the testing sample are unknown to the model. It enables evaluating the effectiveness of the model in conditions similar to those in business practice. In literature, the vast majority of scientists (e.g.: Ooghe & Balcaen, 2006; or Kumar & Ravi, 2007) suggest that the learning dataset was a balanced sample (consisting 50% of entities at risk of bankruptcy, and 50% of entities in good financial condition). This will enable the model to learn to distinguish "good" and "bad" entities. Note, however, that in a market economy the number of firms/consumers at risk of bankruptcy is much smaller than the number of "healthy" entities. Evaluation of the effectiveness of models that use a balanced testing dataset become highly questionable. After all, these models are developed for use in business practice, where the proportion of bankrupts to non-bankrupts is many times smaller. The author of this chapter proved in his previous research that fuzzy logic models are superior over traditional bankruptcy prediction models (both statistical and soft computing techniques) in forecasting risk of bankruptcy of companies in the case of an unbalanced testing dataset (Korol, 2011).

Another controversial aspect on the effectiveness of the most popular analysis methods – multivariate discriminant, logit, and probit , is the possibility of manipulation of the threshold in order to maximize the classification results of these models. This allegation was raised by M. Nwogugu. According to him, the statistical methods do not guarantee reliable results because of the ease at which the threshold which separates "good" and "bad" entities can be manually set (Nwogugu, 2007). Such manipulation, of course, does not increase the effectiveness of the model in business practice after its implementation in a bank, but only in theoretical tests in literature.

Fuzzy Logic in Financial Management 267

arbitrary method of selecting the network architecture. Although there are general formulas to designate the number of hidden neurons, in literature it is postulated to use an individual

To conduct this research1 the author has used the demographical and financial variables of 500 Polish consumers who took consumption credit (400 consumers were "non-bankrupt" – they were repaying the credit with no delays and 100 clients were "bankrupt" – those who had delays in repayment longer than 3 months2). This population of consumers was divided

learning dataset - used for developing the model. There were 50 bankrupt consumers

 testing dataset "one" – used for testing the model created in conditions of an equal proportion of bankrupt and non-bankrupt customers. There were 50 "good" consumers

 testing dataset "two" – consisting of all the customers from testing dataset "one" with the addition of 300 non-bankrupt ones. This enabled testing the ability of the model created to identify customers who have problems with credit repayment among nonbankrupt bank clients in the business practice in proportion of 12,5%/87,5% ("50 bad

All customers were described by 10 demographical and financial variables (Table 2). Additionally, all credit takers were marked with 0-1 variables (0-bankrupt, 1-non-bankrupt).

X4 Number of children in household

X6 Length of employment (in years) X7 Type of employment contract X8 Value of owned car

X9 Net Value of owned apartment/house

X5 Monthly income

X10 Value of other assets

The structure of the developed model is presented in Figure 4. The model consists of four different rule blocks. Rule Block 1 "demographics" evaluates the consumer's demographical

1 All fuzzy logic models were programmed by the author with the use of software – FuzzyTech 5.54d. 2 In Poland at that time there was no law for consumer bankruptcy. Such law was introduced in 2009.

X1 Age X2 Education X3 Marital status

and arbitrary approach for each forecasted phenomenon separately.

and 50 credit applicants in danger of going bankrupt.

Variable Symbol Type of Variable

Table 2. Demographical and Financial Variables of Customers.

**3. Consumer credit scoring model** 

and 50 non-bankrupt ones.

customers"/"350 good ones").

**3.2 Fuzzy logic model** 

**3.1 Research assumptions** 

into:

The next complaint toward traditional bankruptcy models is the issue of becoming obsolete with the passage of time since their estimation. It is assumed that the models function well for 4-6 years, after which it is necessary to modify and update them (Agarwal & Taffler, 2007). It should be noted, however, that the model life cycle presented in Figure 3 is only generally accepted, but there are no strict rules that exactly define the length of the model's life cycle. Forecasting applications become outdated as a result of changes in the business cycle, changing economic conditions which influence the change of appropriate values of financial ratios of the entities (Altman & Rijken, 2006). Fuzzy logic models, of course, also get outdated, but unlike the traditional models, it is easy to update them according to the changing environment without the need for their re-estimation as in the case of statistical models or most of the soft computing techniques.

Fig. 3. The Life Cycle of the Bankruptcy Prediction Model.

In relation to statistical models scientists also mention an allegation about the adoption of an assumption about normal distribution of financial ratios of analyzed companies during the estimating of models (Mcleay & Omar, 2000). This assumption is often not observed due to the fact that few variables are characterized by such distribution. However the desire to meet this assumption, significantly limit the number of indicators that truly reflect the financial situation of the company and thus would cause deterioration in the effectiveness of models of this type.

Artificial neural networks, belonging to the soft computing methods, are not subject to the above drawback concerning the normal distribution of financial ratios. This does not mean that they are free from other defects in predicting the financial situation of companies and consumers. The most common complaint encountered in literature is the inability to justify the decisions made. Often the way artificial neural networks forecast are described as a "black-box system" (Bose & Mahapatra, 2001). Analysis of the process for assigning individual variable weights is complex and difficult to interpret. Neural networks do not provide the course of reasoning leading to certain assessment. They only give their outcome, without being able to trace further evidence leading to a final conclusion. This makes it difficult to correctly identify the causes of generated errors by an artificial neural network. Another drawback of the use of artificial neural networks in predicting bankruptcy is an

The next complaint toward traditional bankruptcy models is the issue of becoming obsolete with the passage of time since their estimation. It is assumed that the models function well for 4-6 years, after which it is necessary to modify and update them (Agarwal & Taffler, 2007). It should be noted, however, that the model life cycle presented in Figure 3 is only generally accepted, but there are no strict rules that exactly define the length of the model's life cycle. Forecasting applications become outdated as a result of changes in the business cycle, changing economic conditions which influence the change of appropriate values of financial ratios of the entities (Altman & Rijken, 2006). Fuzzy logic models, of course, also get outdated, but unlike the traditional models, it is easy to update them according to the changing environment without the need for their re-estimation as in the case of statistical

In relation to statistical models scientists also mention an allegation about the adoption of an assumption about normal distribution of financial ratios of analyzed companies during the estimating of models (Mcleay & Omar, 2000). This assumption is often not observed due to the fact that few variables are characterized by such distribution. However the desire to meet this assumption, significantly limit the number of indicators that truly reflect the financial situation of the company and thus would cause deterioration in the effectiveness of

**2009 2016 2010 2011**

**A well-functioning model**

Artificial neural networks, belonging to the soft computing methods, are not subject to the above drawback concerning the normal distribution of financial ratios. This does not mean that they are free from other defects in predicting the financial situation of companies and consumers. The most common complaint encountered in literature is the inability to justify the decisions made. Often the way artificial neural networks forecast are described as a "black-box system" (Bose & Mahapatra, 2001). Analysis of the process for assigning individual variable weights is complex and difficult to interpret. Neural networks do not provide the course of reasoning leading to certain assessment. They only give their outcome, without being able to trace further evidence leading to a final conclusion. This makes it difficult to correctly identify the causes of generated errors by an artificial neural network. Another drawback of the use of artificial neural networks in predicting bankruptcy is an

models or most of the soft computing techniques.

**pattern pattern today**

Fig. 3. The Life Cycle of the Bankruptcy Prediction Model.

models of this type.

arbitrary method of selecting the network architecture. Although there are general formulas to designate the number of hidden neurons, in literature it is postulated to use an individual and arbitrary approach for each forecasted phenomenon separately.

#### **3. Consumer credit scoring model**

#### **3.1 Research assumptions**

To conduct this research1 the author has used the demographical and financial variables of 500 Polish consumers who took consumption credit (400 consumers were "non-bankrupt" – they were repaying the credit with no delays and 100 clients were "bankrupt" – those who had delays in repayment longer than 3 months2). This population of consumers was divided into:


All customers were described by 10 demographical and financial variables (Table 2). Additionally, all credit takers were marked with 0-1 variables (0-bankrupt, 1-non-bankrupt).


Table 2. Demographical and Financial Variables of Customers.

#### **3.2 Fuzzy logic model**

The structure of the developed model is presented in Figure 4. The model consists of four different rule blocks. Rule Block 1 "demographics" evaluates the consumer's demographical

 1 All fuzzy logic models were programmed by the author with the use of software – FuzzyTech 5.54d.

<sup>2</sup> In Poland at that time there was no law for consumer bankruptcy. Such law was introduced in 2009.

Fuzzy Logic in Financial Management 269

Age (value ranges: 18 years old -65 years old) Young: less than 33

Number of children (value ranges: 0-5 children) Few: less than 2.0

Education level (value ranges: 0-3; where: 0 – elementary education, 1 – high skilled worker, 2 – college education, 3 – university education, doctorate, or high qualified experts)

Marital status (value ranges: 0-1; where: 0 – single, 1- married, between 0 and 1 other types of marital status which can improve financial situation of consumer, e.g.: partnership or widow etc.)

Monthly income (value ranges: 800 PLN – 5000

Length of employment (value ranges: 0 years – 15

Type of employment contract (value ranges: 0-2, where: 0 – agreement on task job, 1 – agreement on limited duration work, 2 – agreement on

Value of car (value ranges: 10 000 PLN – 100 000

Net value of apartment (value ranges: 0 PLN – 500

Value of other assets (value ranges: 1000 PLN – 20

for Rule Block 1 "Demographics":

Table 3. Threshold Values for Membership Functions of Entry Variables

If X1 is Young and X2 is Basic and X3 is Single and X4 is Few then Demographics is Weak If X1 is Young and X2 is Average and X3 is Single and X4 is Few then Demographics is Average

The exemplary form of the membership functions are presented in Figure 5 for the variable - "Age" and in Figure 6 for variable - "Output". Following set of decision rules was created

PLN)

years)

PLN)

000 PLN)

000 PLN)

indefinite duration job)

**Variable Criteria (thresholds for individual** 

**membership functions)** 

Middle age: from 27 to 53 Old: more than 48

Average: from 1.0 to 3.7 Many: more than 3.0

Basic level: less than 1.0 Average level: from 0.8 to 2.25 High level: more than 1.5

Single: less than 0.7 Married: more than 0.7

Short: less than 7.5 Medium: from 3.7 to 11.25 Long: more than 7.5

PLN

PLN

PLN

Low income: less than 2900 PLN

High income: more than 2950 PLN

Only task job – less than 1.0

Cheap: less than 55 000 PLN

Low: less than 325 000 PLN

High: more than 325 000 PLN

High: more than 10 500 PLN

Low: less than 4500 PLN

Average income: from 1850 PLN to 3950

Limited duration work – from 0.5 to 1.5 Indefinite duration job – more than 1.0

Middle class: from 30 000 PLN to 77 500

Average: from 237 500 PLN to 412 500

Average: from 2700 PLN to 15 250 PLN

Expensive: more than 55 000 PLN

variables (age, education level, marital status, number of children in household). Rule Block 2 "finance" assesses the financial condition of the consumer based on three variables (monthly income, the length of employment, type of employment contract). Rule Block 3 "financial security" analyzes the financial strength of the customer and eventually the security for the granted credit. Rule Block 4 "the score" uses as entry variables the forecasted output of all three Rule Blocks, which are: demographics variable (there are three states of demographics forecasted at Rule Block 1: weak, average, strong), finance variable (there are three states of financial strength forecasted in Rule Block 2: weak, average, strong), and financial security variable (there are three states of security forecasted at Rule Block 3: weak, average, strong). Based on these three evaluated inputs the model forecasts the final credit scoring output.

The model's output is a variable representing a forecast of the financial situation of an audited consumer. This variable ranges from 0 to 1, while it is assumed that there are three levels of risk: high risk for values smaller than 0.3, medium risk for values from 0.3 to 0.7, and low risk for values larger than 0.7.

Fig. 4. Structure of the Fuzzy Logic Model for Consumer Credit Scoring.

This model is based on sets of rules written by the author in the form of IF - THEN, where expert knowledge is stored. For each entry variable to the model, the author identified from two to three fuzzy sets (which are subsets of a set of values of the entry variable), and their corresponding membership functions. The fuzzy sets and the shape of membership functions have been arbitrarily designated by the author. The fuzzy sets and the thresholds for all membership functions are presented in Table 3.

variables (age, education level, marital status, number of children in household). Rule Block 2 "finance" assesses the financial condition of the consumer based on three variables (monthly income, the length of employment, type of employment contract). Rule Block 3 "financial security" analyzes the financial strength of the customer and eventually the security for the granted credit. Rule Block 4 "the score" uses as entry variables the forecasted output of all three Rule Blocks, which are: demographics variable (there are three states of demographics forecasted at Rule Block 1: weak, average, strong), finance variable (there are three states of financial strength forecasted in Rule Block 2: weak, average, strong), and financial security variable (there are three states of security forecasted at Rule Block 3: weak, average, strong). Based on these three evaluated inputs the model forecasts

The model's output is a variable representing a forecast of the financial situation of an audited consumer. This variable ranges from 0 to 1, while it is assumed that there are three levels of risk: high risk for values smaller than 0.3, medium risk for values from 0.3 to 0.7,

Fig. 4. Structure of the Fuzzy Logic Model for Consumer Credit Scoring.

for all membership functions are presented in Table 3.

This model is based on sets of rules written by the author in the form of IF - THEN, where expert knowledge is stored. For each entry variable to the model, the author identified from two to three fuzzy sets (which are subsets of a set of values of the entry variable), and their corresponding membership functions. The fuzzy sets and the shape of membership functions have been arbitrarily designated by the author. The fuzzy sets and the thresholds

the final credit scoring output.

and low risk for values larger than 0.7.


Table 3. Threshold Values for Membership Functions of Entry Variables

The exemplary form of the membership functions are presented in Figure 5 for the variable - "Age" and in Figure 6 for variable - "Output". Following set of decision rules was created for Rule Block 1 "Demographics":

If X1 is Young and X2 is Basic and X3 is Single and X4 is Few then Demographics is Weak If X1 is Young and X2 is Average and X3 is Single and X4 is Few then Demographics is Average

Fuzzy Logic in Financial Management 271

Fig. 5. Defined Membership Functions of Variable "Age".

Fig. 6. Membership Functions of Variable "Output".

Based on above set of decision rules the model evaluates a consumer's demographical situation that has direct influence on their credibility. There are four variables analyzed in this rule block: age of consumer, education level, marital status, and number of children in household. The rules are constructed in such a way to consider the different influence each variable has on the strength of a consumer's demographical state. Level of education (values from 0 to 3) is considered to have a positive influence on the credibility of the credit

If X1 is Young and X2 is High and X3 is Single and X4 is Few then Demographics is Average If X1 is Middle age and X2 is Basic and X3 is Single and X4 is Few then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Single and X4 is Few then Demographics is Average If X1 is Middle age and X2 is High and X3 is Single and X4 is Few then Demographics is Average If X1 is Old and X2 is Basic and X3 is Single and X4 is Few then Demographics is Weak If X1 is Old and X2 is Average and X3 is Single and X4 is Few then Demographics is Average If X1 is Old and X2 is High and X3 is Single and X4 is Few then Demographics is Average If X1 is Young and X2 is Basic and X3 is Married and X4 is Few then Demographics is Weak If X1 is Young and X2 is Average and X3 is Married and X4 is Few then Demographics is Average If X1 is Young and X2 is High and X3 is Married and X4 is Few then Demographics is Strong If X1 is Middle age and X2 is Basic and X3 is Married and X4 is Few then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Married and X4 is Few then Demographics is Average If X1 is Middle age and X2 is High and X3 is Married and X4 is Few then Demographics is Strong If X1 is Old and X2 is Basic and X3 is Married and X4 is Few then Demographics is Weak If X1 is Old and X2 is Average and X3 is Married and X4 is Few then Demographics is Average If X1 is Old and X2 is High and X3 is Married and X4 is Few then Demographics is Strong If X1 is Young and X2 is Basic and X3 is Single and X4 is Average then Demographics is Weak If X1 is Young and X2 is Average and X3 is Single and X4 is Average then Demographics is Weak If X1 is Young and X2 is High and X3 is Single and X4 is Average then Demographics is Average If X1 is Middle age and X2 is Basic and X3 is Single and X4 is Average then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Single and X4 is Average then Demographics is Average If X1 is Middle age and X2 is High and X3 is Single and X4 is Average then Demographics is Average If X1 is Old and X2 is Basic and X3 is Single and X4 is Average then Demographics is Weak If X1 is Old and X2 is Average and X3 is Single and X4 is Average then Demographics is Average If X1 is Old and X2 is High and X3 is Single and X4 is Average then Demographics is Average If X1 is Young and X2 is Basic and X3 is Married and X4 is Average then Demographics is Weak If X1 is Young and X2 is Average and X3 is Married and X4 is Average then Demographics is Average If X1 is Young and X2 is High and X3 is Married and X4 is Average then Demographics is Strong If X1 is Middle age and X2 is Basic and X3 is Married and X4 is Average then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Married and X4 is Average then Demographics is Average If X1 is Middle age and X2 is High and X3 is Married and X4 is Average then Demographics is Strong If X1 is Old and X2 is Basic and X3 is Married and X4 is Average then Demographics is Weak If X1 is Old and X2 is Average and X3 is Married and X4 is Average then Demographics is Average If X1 is Old and X2 is High and X3 is Married and X4 is Average then Demographics is Strong If X1 is Young and X2 is Basic and X3 is Single and X4 is Many then Demographics is Weak If X1 is Young and X2 is Average and X3 is Single and X4 is Many then Demographics is Weak If X1 is Young and X2 is High and X3 is Single and X4 is Many then Demographics is Average If X1 is Middle age and X2 is Basic and X3 is Single and X4 is Many then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Single and X4 is Many then Demographics is Average If X1 is Middle age and X2 is High and X3 is Single and X4 is Many then Demographics is Average If X1 is Old and X2 is Basic and X3 is Single and X4 is Many then Demographics is Weak If X1 is Old and X2 is Average and X3 is Single and X4 is Many then Demographics is Weak If X1 is Old and X2 is High and X3 is Single and X4 is Many then Demographics is Average If X1 is Young and X2 is Basic and X3 is Married and X4 is Many then Demographics is Weak If X1 is Young and X2 is Average and X3 is Married and X4 is Many then Demographics is Average If X1 is Young and X2 is High and X3 is Married and X4 is Many then Demographics is Average If X1 is Middle age and X2 is Basic and X3 is Married and X4 is Many then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Married and X4 is Many then Demographics is Average If X1 is Middle age and X2 is High and X3 is Married and X4 is Many then Demographics is Average If X1 is Old and X2 is Basic and X3 is Married and X4 is Many then Demographics is Weak If X1 is Old and X2 is Average and X3 is Married and X4 is Many then Demographics is Average If X1 is Old and X2 is High and X3 is Married and X4 is Many then Demographics is Average

If X1 is Young and X2 is High and X3 is Single and X4 is Few then Demographics is Average If X1 is Middle age and X2 is Basic and X3 is Single and X4 is Few then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Single and X4 is Few then Demographics is Average If X1 is Middle age and X2 is High and X3 is Single and X4 is Few then Demographics is Average

If X1 is Old and X2 is Basic and X3 is Single and X4 is Few then Demographics is Weak If X1 is Old and X2 is Average and X3 is Single and X4 is Few then Demographics is Average If X1 is Old and X2 is High and X3 is Single and X4 is Few then Demographics is Average If X1 is Young and X2 is Basic and X3 is Married and X4 is Few then Demographics is Weak If X1 is Young and X2 is Average and X3 is Married and X4 is Few then Demographics is Average If X1 is Young and X2 is High and X3 is Married and X4 is Few then Demographics is Strong If X1 is Middle age and X2 is Basic and X3 is Married and X4 is Few then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Married and X4 is Few then Demographics is Average If X1 is Middle age and X2 is High and X3 is Married and X4 is Few then Demographics is Strong If X1 is Old and X2 is Basic and X3 is Married and X4 is Few then Demographics is Weak If X1 is Old and X2 is Average and X3 is Married and X4 is Few then Demographics is Average If X1 is Old and X2 is High and X3 is Married and X4 is Few then Demographics is Strong If X1 is Young and X2 is Basic and X3 is Single and X4 is Average then Demographics is Weak If X1 is Young and X2 is Average and X3 is Single and X4 is Average then Demographics is Weak If X1 is Young and X2 is High and X3 is Single and X4 is Average then Demographics is Average If X1 is Middle age and X2 is Basic and X3 is Single and X4 is Average then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Single and X4 is Average then Demographics is Average If X1 is Middle age and X2 is High and X3 is Single and X4 is Average then Demographics is Average

If X1 is Old and X2 is Basic and X3 is Single and X4 is Average then Demographics is Weak If X1 is Old and X2 is Average and X3 is Single and X4 is Average then Demographics is Average If X1 is Old and X2 is High and X3 is Single and X4 is Average then Demographics is Average If X1 is Young and X2 is Basic and X3 is Married and X4 is Average then Demographics is Weak If X1 is Young and X2 is Average and X3 is Married and X4 is Average then Demographics is Average If X1 is Young and X2 is High and X3 is Married and X4 is Average then Demographics is Strong If X1 is Middle age and X2 is Basic and X3 is Married and X4 is Average then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Married and X4 is Average then Demographics is Average If X1 is Middle age and X2 is High and X3 is Married and X4 is Average then Demographics is Strong If X1 is Old and X2 is Basic and X3 is Married and X4 is Average then Demographics is Weak If X1 is Old and X2 is Average and X3 is Married and X4 is Average then Demographics is Average If X1 is Old and X2 is High and X3 is Married and X4 is Average then Demographics is Strong If X1 is Young and X2 is Basic and X3 is Single and X4 is Many then Demographics is Weak If X1 is Young and X2 is Average and X3 is Single and X4 is Many then Demographics is Weak If X1 is Young and X2 is High and X3 is Single and X4 is Many then Demographics is Average If X1 is Middle age and X2 is Basic and X3 is Single and X4 is Many then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Single and X4 is Many then Demographics is Average If X1 is Middle age and X2 is High and X3 is Single and X4 is Many then Demographics is Average

If X1 is Old and X2 is Basic and X3 is Single and X4 is Many then Demographics is Weak If X1 is Old and X2 is Average and X3 is Single and X4 is Many then Demographics is Weak If X1 is Old and X2 is High and X3 is Single and X4 is Many then Demographics is Average If X1 is Young and X2 is Basic and X3 is Married and X4 is Many then Demographics is Weak If X1 is Young and X2 is Average and X3 is Married and X4 is Many then Demographics is Average If X1 is Young and X2 is High and X3 is Married and X4 is Many then Demographics is Average If X1 is Middle age and X2 is Basic and X3 is Married and X4 is Many then Demographics is Weak If X1 is Middle age and X2 is Average and X3 is Married and X4 is Many then Demographics is Average If X1 is Middle age and X2 is High and X3 is Married and X4 is Many then Demographics is Average

If X1 is Old and X2 is Basic and X3 is Married and X4 is Many then Demographics is Weak If X1 is Old and X2 is Average and X3 is Married and X4 is Many then Demographics is Average If X1 is Old and X2 is High and X3 is Married and X4 is Many then Demographics is Average

Fig. 5. Defined Membership Functions of Variable "Age".

Fig. 6. Membership Functions of Variable "Output".

Based on above set of decision rules the model evaluates a consumer's demographical situation that has direct influence on their credibility. There are four variables analyzed in this rule block: age of consumer, education level, marital status, and number of children in household. The rules are constructed in such a way to consider the different influence each variable has on the strength of a consumer's demographical state. Level of education (values from 0 to 3) is considered to have a positive influence on the credibility of the credit

Fuzzy Logic in Financial Management 273

applicant (the higher level of education the better). In the same positive way marital status (values from 0 to 1) affects the output of Rule Block 1. However, number of children in household (values from 0 to 5) has a negative influence on a consumer's status. A client's age in certain values (range of values for the middle aged category) has a positive affect on the output, and in other cases negatively influences the score (range of values for the young

The complete block diagram containing all set of decision rules for created Rule Block 2 "Finance", Rule Block 3 "Financial Security" is presented in table 4, and for the output Rule Block 4 "The score" is presented in table 5 (the variables are described in table 2 and 4).

Rule Block 4 "The Score"

Weak Weak Weak High risk Weak Weak Average High risk Weak Weak Strong High risk Weak Average Weak High risk Weak Average Average Medium risk Weak Average Strong Medium risk Weak Strong Weak Medium risk Weak Strong Average Medium risk Weak Strong Strong Low risk Average Weak Weak High risk Average Weak Average High risk Average Weak Strong Medium risk Average Average Weak Medium risk Average Average Average Medium risk Average Average Strong Medium risk Average Strong Weak Medium risk Average Strong Average Low risk Average Strong Strong Low risk Strong Weak Weak Medium risk Strong Weak Average Medium risk Strong Weak Strong Medium risk Strong Average Weak Medium risk Strong Average Average Medium risk Strong Average Strong Low risk Strong Strong Weak Low risk Strong Strong Average Low risk Strong Strong Strong Low risk

If "Financial Security" is:

Then final output of the model "The Score" is:

If "Finance" is:

Table 5. The Set of Decision Rules for Rule Block 4

and old category).

If "Demographics" is:


Table 4. The Set of Decision Rules for Rule Block 2 and Rule Block 3

Low Short Task job Weak Cheap Low Low Weak Low Medium Task job Weak Cheap Average Low Weak Low Long Task job Average Cheap High Low Average Low Short Limited dur. Weak Cheap Low Average Weak Low Medium Limited dur. Weak Cheap Average Average Weak Low Long Limited dur. Average Cheap High Average Strong

Average Short Task job Weak Middle class Low Low Weak Average Medium Task job Average Middle class Average Low Average Average Long Task job Average Middle class High Low Strong Average Short Limited dur. Weak Middle class Low Average Average Average Medium Limited dur. Average Middle class Average Average Average Average Long Limited dur. Average Middle class High Average Strong

High Short Task job Average Expensive Low Low Weak High Medium Task job Average Expensive Average Low Average High Long Task job Strong Expensive High Low Strong High Short Limited dur. Average Expensive Low Average Average High Medium Limited dur. Strong Expensive Average Average Average High Long Limited dur. Strong Expensive High Average Strong

Then output "Finance" is:

If X5 is: If X6 is: If X7 is:

Low Short Indefinite

Low Medium Indefinite

Low Long Indefinite

Average Short Indefinite

Average Medium Indefinite

Average Long Indefinite

High Short Indefinite

High Medium Indefinite

High Long Indefinite

Table 4. The Set of Decision Rules for Rule Block 2 and Rule Block 3

Rule Block 2 "Finance" Rule Block 3 "Financial Security"

If X8 is: If X9 is: If X10 is

dur. Weak Cheap Low High Weak

dur. Average Cheap Average High Average

dur. Average Cheap High High Strong

dur. Average Middle class Low High Average

dur. Average Middle class Average High Average

dur. Strong Middle class High High Strong

dur. Strong Expensive Low High Average

dur. Strong Expensive Average High Strong

dur. Strong Expensive High High Strong

Then output "Financial Security" is:

applicant (the higher level of education the better). In the same positive way marital status (values from 0 to 1) affects the output of Rule Block 1. However, number of children in household (values from 0 to 5) has a negative influence on a consumer's status. A client's age in certain values (range of values for the middle aged category) has a positive affect on the output, and in other cases negatively influences the score (range of values for the young and old category).

The complete block diagram containing all set of decision rules for created Rule Block 2 "Finance", Rule Block 3 "Financial Security" is presented in table 4, and for the output Rule Block 4 "The score" is presented in table 5 (the variables are described in table 2 and 4).


Table 5. The Set of Decision Rules for Rule Block 4

Fuzzy Logic in Financial Management 275



It is necessary to make a note that a I type error is much more costly than a II type error to make. I type error means that a bank classifies a bankrupt consumer as a non-bankrupt one.

The results obtained from testing the developed model against the bankruptcy risk while

Table 6. Results of Effectiveness of the Fuzzy Logic Model in Consumer Credit Scoring

operations. The overall effectiveness of this model obtained from that dataset was 91%.

by author, it can be seen that author's model is characterized by:

In the case of testing dataset "one", it can be seen that the fuzzy logic model created evaluated 9 credit applications incorrectly. Among those, 5 cases concerned classification of consumers with the risk of insolvency as "good" borrowers, and remaining 4 mistakes where II type errors, which means that the model classified "good" credit applicants as the high risk

Due to the equal distribution of "bad" and "good" consumers in testing dataset "one", the author treats this research approach as a theoretical possibility test of the predictive power of the method used. From the viewpoint of the practical applicability of the fuzzy logic model in business, the conclusions from the tests conducted on testing dataset "two", which contained 87.5% consumers with good financial condition and 12.5% consumers at risk of insolvency, are more important to analyze. When testing the model with such a proportion of "bad" and "good" consumers, the II type mistakes increased by 3.43 percentage points (from 8% to 11.42%). This caused the decrease of overall effectiveness of the model from 91% to 88.75%. Nevertheless such effectiveness can be rated as high. Unlike the models predicting bankruptcy of firms, it is difficult to conduct comparative analysis of effectiveness of models forecasting bankruptcy of consumers. Models used in literature are theoretical ones, or their authors do not provide results, or they are models for commercial use of restricted character. From the available research, the results of statistical models vary from 72 % (Tingting, 2006) to 77.5 % (Boyle et al., 1992) – figure 7. Comparing the overall effectiveness of the models found in literature to effectiveness of fuzzy logic model created

**Testing Type Effectiveness** 

E1 10% (5 cases) E2 8% (4 cases) S 91%

E1 10% (5 cases) E2 11.42% (40 cases) S 88.75%


testing dataset "one" and "two" are presented in Table 6.

Testing dataset "One" (50 "bad" / 50 "good" consumers)

Testing dataset "Two" (50 "bad" / 350 "good" consumers)

II type error means that a non-bankrupt entity is classified as a bankrupt one.

the testing set;

testing set;

In the Rule Block 2 "Finance" there are three variables analyzed: monthly income, the length of employment and the type of employment contract. Based on set of decision rules in this rule block, the model evaluates a consumer's financial strength that has influence on their credibility. It is considered that each variable has different influence on the financial strength of the customer. Monthly income (values from 800 PLN to 5000 PLN) and length of employment (values from 0 to 15 years) are considered to have a positive influence on the financial stability of the customer (the higher value the better). Third variable – the type of employment contract, defines if the customer source of monthly income is stable. There are three types of the contracts specified: task job contract, limited duration contract, indefinite duration contract. The task job contract is considered to be the worst for the stability of the customer's income. The best contract is indefinite duration one.

In case of Rule Block 3 "Financial Security" there are following three variables analyzed: value of the car, net value of the apartment/house, value of other assets. The task of this rule block is to evaluate the loan collateral. The rules are constructed in such a way to analyze the positive influence of all three variables on financial security of the customer. In addition the net value of apartment/house is considered to have dominant role on the output of this rule block, as it is characterized by the highest value and stability than two other variables.

The outputs of rule blocks 1, 2, and 3 are considered as input variables to the Rule Block 4 "The Score". The model's output "The Score" is a variable representing a forecast of the financial situation of an audited consumer. As it was mentioned earlier in this section of chapter, the output variable ranges from 0 to 1, while it is assumed that there are three levels of risk: high risk for values smaller than 0.3, medium risk for values from 0.3 to 0.7, and low risk for values larger than 0.7.

The use of variables (financial and demographical – Figure 4) implemented in this research is consistent with the credit scoring applications in literature. Most authors mainly use age, education, employment/unemployment status, monthly income, and number of children in household in consumer credit scoring models (e.g.: Henley & Hand, 1996; Wiginton, 1980; Thomas, 2000; Tingting, 2006). As described in Section 2 of this chapter, most of the credit scoring applications are statistical models. One of the newest examples of a developed model is the probit model with nine variables (Tingting, 2006). The estimates for each variable in this model are as follows: if consumer was unemployed (1.4207), family income in \$00,000 (-0.155), state property exemption in \$0,000 (0.1802), if consumer is collegeeducated (-0.4677), age of consumer (-0.1541), if consumer is male (-0.3354), if consumer is married (-0.0693), if consumer is white (-0.1838), number of children (0.0401). The variables with negative estimates positively influence the risk of bankruptcy (the higher variable value the lower risk of going bankrupt) and variables with positive estimates negatively influence the risk of insolvency (the higher variable value the higher risk). From the form of the model it can be seen that education and status of employment were influencing the output of the model the most3.

#### **3.3 The results**

Model was evaluated based on two types of errors and overall effectiveness:

<sup>3</sup> In the Tingting (2006) paper a few of the variables used seem controversial (e.g. taking the sex or race of a consumer under consideration in the credit scoring procedure).

In the Rule Block 2 "Finance" there are three variables analyzed: monthly income, the length of employment and the type of employment contract. Based on set of decision rules in this rule block, the model evaluates a consumer's financial strength that has influence on their credibility. It is considered that each variable has different influence on the financial strength of the customer. Monthly income (values from 800 PLN to 5000 PLN) and length of employment (values from 0 to 15 years) are considered to have a positive influence on the financial stability of the customer (the higher value the better). Third variable – the type of employment contract, defines if the customer source of monthly income is stable. There are three types of the contracts specified: task job contract, limited duration contract, indefinite duration contract. The task job contract is considered to be the worst for the stability of the

In case of Rule Block 3 "Financial Security" there are following three variables analyzed: value of the car, net value of the apartment/house, value of other assets. The task of this rule block is to evaluate the loan collateral. The rules are constructed in such a way to analyze the positive influence of all three variables on financial security of the customer. In addition the net value of apartment/house is considered to have dominant role on the output of this rule block, as it is characterized by the highest value and stability than two other variables. The outputs of rule blocks 1, 2, and 3 are considered as input variables to the Rule Block 4 "The Score". The model's output "The Score" is a variable representing a forecast of the financial situation of an audited consumer. As it was mentioned earlier in this section of chapter, the output variable ranges from 0 to 1, while it is assumed that there are three levels of risk: high risk for values smaller than 0.3, medium risk for values from 0.3 to 0.7, and low

The use of variables (financial and demographical – Figure 4) implemented in this research is consistent with the credit scoring applications in literature. Most authors mainly use age, education, employment/unemployment status, monthly income, and number of children in household in consumer credit scoring models (e.g.: Henley & Hand, 1996; Wiginton, 1980; Thomas, 2000; Tingting, 2006). As described in Section 2 of this chapter, most of the credit scoring applications are statistical models. One of the newest examples of a developed model is the probit model with nine variables (Tingting, 2006). The estimates for each variable in this model are as follows: if consumer was unemployed (1.4207), family income in \$00,000 (-0.155), state property exemption in \$0,000 (0.1802), if consumer is collegeeducated (-0.4677), age of consumer (-0.1541), if consumer is male (-0.3354), if consumer is married (-0.0693), if consumer is white (-0.1838), number of children (0.0401). The variables with negative estimates positively influence the risk of bankruptcy (the higher variable value the lower risk of going bankrupt) and variables with positive estimates negatively influence the risk of insolvency (the higher variable value the higher risk). From the form of the model it can be seen that education and status of employment were influencing the

Model was evaluated based on two types of errors and overall effectiveness:

of a consumer under consideration in the credit scoring procedure).

3 In the Tingting (2006) paper a few of the variables used seem controversial (e.g. taking the sex or race

customer's income. The best contract is indefinite duration one.

risk for values larger than 0.7.

output of the model the most3.

**3.3 The results** 


It is necessary to make a note that a I type error is much more costly than a II type error to make. I type error means that a bank classifies a bankrupt consumer as a non-bankrupt one. II type error means that a non-bankrupt entity is classified as a bankrupt one.

The results obtained from testing the developed model against the bankruptcy risk while testing dataset "one" and "two" are presented in Table 6.


Table 6. Results of Effectiveness of the Fuzzy Logic Model in Consumer Credit Scoring

In the case of testing dataset "one", it can be seen that the fuzzy logic model created evaluated 9 credit applications incorrectly. Among those, 5 cases concerned classification of consumers with the risk of insolvency as "good" borrowers, and remaining 4 mistakes where II type errors, which means that the model classified "good" credit applicants as the high risk operations. The overall effectiveness of this model obtained from that dataset was 91%.

Due to the equal distribution of "bad" and "good" consumers in testing dataset "one", the author treats this research approach as a theoretical possibility test of the predictive power of the method used. From the viewpoint of the practical applicability of the fuzzy logic model in business, the conclusions from the tests conducted on testing dataset "two", which contained 87.5% consumers with good financial condition and 12.5% consumers at risk of insolvency, are more important to analyze. When testing the model with such a proportion of "bad" and "good" consumers, the II type mistakes increased by 3.43 percentage points (from 8% to 11.42%). This caused the decrease of overall effectiveness of the model from 91% to 88.75%. Nevertheless such effectiveness can be rated as high. Unlike the models predicting bankruptcy of firms, it is difficult to conduct comparative analysis of effectiveness of models forecasting bankruptcy of consumers. Models used in literature are theoretical ones, or their authors do not provide results, or they are models for commercial use of restricted character. From the available research, the results of statistical models vary from 72 % (Tingting, 2006) to 77.5 % (Boyle et al., 1992) – figure 7. Comparing the overall effectiveness of the models found in literature to effectiveness of fuzzy logic model created by author, it can be seen that author's model is characterized by:

Fuzzy Logic in Financial Management 277

 learning dataset - used for developing the models. There were 25 bankrupt companies and 28 non-bankrupt ones. Those 53 companies were from various sectors such as construction, metal industry, food processing, chemicals, telecommunications, etc. testing dataset "one" – used for testing the models created in conditions of an equal proportion of bankrupt and non-bankrupt firms. There were 29 "healthy" firms and 25

 testing dataset "two" – consisting of all the companies from testing dataset "one" with the addition of 78 non-bankrupt companies. This enabled testing the ability of the models created to identify bankrupt companies among non-bankrupt firms in the business practice in the proportion of 19%/81% ("25 bad enterprises"/"107 good

All models were tested by testing dataset "one" and "two" for both two years prior to

All companies were described by 14 calculated financial ratios for two years before bankruptcy. These ratios are presented in Table 7. Additionally, all firms were marked with 0-1 variables (0-bankrupt, 1-non-bankrupt). Both models were evaluated based on two types of errors and overall effectiveness using the same formulas as in the previous section of the

**PROFITABILITY RATIOS** 

**LIQUIDITY RATIOS** 

**DEBT RATIOS** 

**X8** (net profit + amortization) / Long term and short term liabilities

**X11** (Stockholders equity + long term liabilities) / fixed assets **ACTIVITY RATIOS** 

**OTHER RATIOS** 

Before programming the bankruptcy prediction models for both years prior to the insolvency of firms with the use of both methods (fuzzy logic and artificial neural

**X4** [Current assets - inventories] / short term liabilities

**Ratio Symbol Type of Ratio and Calculation Formula** 

**X1** Profit from sales / total assets **X2** Operating profit / revenues from sales

**X3** Current assets / short term liabilities

**X5** Working capital / total assets

**X6** Short term liabilities / total assets **X7** Equity / total credits

**X10** Gross profit / short term liabilities

**X9** Operating costs / short term liabilities **X12** Net revenues / total assets **X13** Net revenues / short term receivables

**X14** Log of total assets

Table 7. Financial Ratios Used in the Research

**4.2 Early warning models for enterprises** 

companies in danger of going bankrupt.

enterprises").

bankruptcy.

chapter.


Fig. 7. The Comparison of Effectiveness of Fuzzy Logic Models

#### **4. Business credit scoring model**

#### **4.1 Research assumptions**

The author of this chapter has created 2 fuzzy logic models in order to verify the influence of the following aspects on the quality of the forecast:


To conduct this research the author has used the financial statements of 185 Polish stock equity companies (135 non-bankrupt and 50 bankrupt) from the years 2000-2007. This population of firms was divided into:





The author of this chapter has created 2 fuzzy logic models in order to verify the influence of

ability of a fuzzy logic model to predict bankruptcy of companies for one year, two

 comparison of the effectiveness of fuzzy logic with the most popular form among artificial intelligence methods – neural network model, and with the effectiveness of the first bankruptcy model of Altman created in 1968, which is still the most popular and

To conduct this research the author has used the financial statements of 185 Polish stock equity companies (135 non-bankrupt and 50 bankrupt) from the years 2000-2007. This

proportion of bankrupt and non-bankrupt companies in a testing setdata,

testing sample – 91% vs 72%),

testing sample - 91% vs 77.5%),

testing sample – 88.75% vs 77.5%).

unbalanced testing sample – 88.75% vs 72%),

Fig. 7. The Comparison of Effectiveness of Fuzzy Logic Models

**4. Business credit scoring model** 

the following aspects on the quality of the forecast:

widely used in the business world.

population of firms was divided into:

**4.1 Research assumptions** 

years before,


All models were tested by testing dataset "one" and "two" for both two years prior to bankruptcy.

All companies were described by 14 calculated financial ratios for two years before bankruptcy. These ratios are presented in Table 7. Additionally, all firms were marked with 0-1 variables (0-bankrupt, 1-non-bankrupt). Both models were evaluated based on two types of errors and overall effectiveness using the same formulas as in the previous section of the chapter.


Table 7. Financial Ratios Used in the Research

#### **4.2 Early warning models for enterprises**

Before programming the bankruptcy prediction models for both years prior to the insolvency of firms with the use of both methods (fuzzy logic and artificial neural

Fuzzy Logic in Financial Management 279

assumed that the threshold value separating the "good" and "bad" companies is 0.5 (output variable values below 0,5 mean the company is at risk of bankruptcy, while those above 0.5 represent a company safe from bankruptcy). The final result generated by the fuzzy logic model is based on an assessment of four (one year analysis) and five financial ratios (two years analysis). The rule block in the model consists following set of rules for forecasting the

A set of rules for forecasting the economic situation of companies in two years prior to

If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 > 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1 If X1\_2 > 0.02 and X5\_2 > 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 <= 1.4 and X7\_2 > 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 > 0.8 then 0 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 > 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 1 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 > 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1 If X1\_2 > 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1 If X1\_2 > 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 > 0.8 then 1 If X1\_2 > 0.02 and X5\_2 > 0.14 and X8\_2 > 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 1 If X1\_2 > 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 1 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1

economic situation in one year prior financial failure:

bankruptcy is as follows:

If X3\_1 <= 1.025 and X8\_1 <= 0.03 and X9\_1 <= 2 and X10\_1 <= (-0.1) then 0 If X3\_1 <= 1.025 and X8\_1 <= 0.03 and X9\_1 <= 2 and X10\_1 > (-0.1) then 0 If X3\_1 <=1.025 and X8\_1 <= 0.03 and X9\_1 > 2 and X10\_1 > (-0.1) then 0 If X3\_1 <=1.025 and X8\_1 > 0.03 and X9\_1 > 2 and X10\_1 > (-0.1) then 1 If X3\_1 > 1.025 and X8\_1 > 0.03 and X9\_1 > 2 and X10\_1 > (-0.1) then 1 If X3\_1 > 1.025 and X8\_1 > 0.03 and X9\_1 > 2 and X10\_1 <= (-0.1) then 1 If X3\_1 > 1.025 and X8\_1 > 0.03 and X9\_1 <= 2 and X10\_1 <= (-0.1) then 1 If X3\_1 > 1.025 and X8\_1 <= 0.03 and X9\_1 <= 2 and X10\_1 <= (-0.1) then 0 If X3\_1 <=1.025 and X8\_1 > 0.03 and X9\_1 <= 2 and X10\_1 <= (-0.1) then 0 If X3\_1 <=1.025 and X8\_1 <= 0.03 and X9\_1 > 2 and X10\_1 <= (-0.1) then 0 If X3\_1 <=1.025 and X8\_1 > 0.03 and X9\_1 > 2 and X10\_1 <= (-0.1) then 0 If X3\_1 > 1.025 and X8\_1 <= 0.03 and X9\_1 > 2 and X10\_1 > (-0.1) then 1 If X3\_1 > 1.025 and X8\_1 > 0.03 and X9\_1 <= 2 and X10\_1 > (-0.1) then 1 If X3\_1 > 1.025 and X8\_1 <= 0.03 and X9\_1 <= 2 and X10\_1 > (-0.1) then 0 If X3\_1 > 1.025 and X8\_1 <= 0.03 and X9\_1 > 2 and X10\_1 <= (-0.1) then 0 If X3\_1 <=1.025 and X8\_1 > 0.03 and X9\_1 <= 2 and X10\_1 > (-0.1) then 0

networks), the author conducted a correlation analysis for all ratios from Table 7. The objective of this analysis was to choose ratios that were highly correlated with the score and at the same time had a low correlation between each other. The following ratios were taken into the models as entry data nodes:


For each entry variable to the model, the author identified two fuzzy sets (which are subsets of a set of values of the entry variable): "positive" and "negative", and their corresponding membership functions. The fuzzy sets and the shape of membership functions have been arbitrarily designated by the author.

In order to set the critical values for membership functions in the models, the author calculated for all ratios the first and the third quartile, and median value separately for "good" and "bad" companies. The value of the third quartile of the "bad" firms was used as the threshold value for membership functions. These values are presented in Table 8.


Table 8. The Threshold Values for Membership Functions Used in Both Fuzzy Logic Models.

The set of rules used by the fuzzy decision model contains 16 rules for analysis of companies one year prior to bankruptcy and 25 rules for analysis with an increased period of forecast. Extending the length of prediction to two years prior to insolvency required supporting the models (both fuzzy logic and artificial neural networks) with a larger amount of financial information, i.e. financial ratios.

The structure of the fuzzy logic model created for one year and two years prior to bankruptcy is presented in Figure 8 (Figure 8 presents the use of financial ratios for two years analysis, in case of one year analysis the structure is the same, but used financial ratios are different – see table 8). The model consists of four inputs (financial ratios) in one year prior bankruptcy, five inputs (financial ratios) in two years prior financial failure and one rule block in both years. The model's output is a variable representing a forecast of the financial situation of an audited company. This variable ranges from 0 to 1, while it is

networks), the author conducted a correlation analysis for all ratios from Table 7. The objective of this analysis was to choose ratios that were highly correlated with the score and at the same time had a low correlation between each other. The following ratios were taken

For each entry variable to the model, the author identified two fuzzy sets (which are subsets of a set of values of the entry variable): "positive" and "negative", and their corresponding membership functions. The fuzzy sets and the shape of membership functions have been

In order to set the critical values for membership functions in the models, the author calculated for all ratios the first and the third quartile, and median value separately for "good" and "bad" companies. The value of the third quartile of the "bad" firms was used as

One year prior to bankruptcy

Two years prior to bankruptcy

Table 8. The Threshold Values for Membership Functions Used in Both Fuzzy Logic Models.

The set of rules used by the fuzzy decision model contains 16 rules for analysis of companies one year prior to bankruptcy and 25 rules for analysis with an increased period of forecast. Extending the length of prediction to two years prior to insolvency required supporting the models (both fuzzy logic and artificial neural networks) with a larger amount of financial

The structure of the fuzzy logic model created for one year and two years prior to bankruptcy is presented in Figure 8 (Figure 8 presents the use of financial ratios for two years analysis, in case of one year analysis the structure is the same, but used financial ratios are different – see table 8). The model consists of four inputs (financial ratios) in one year prior bankruptcy, five inputs (financial ratios) in two years prior financial failure and one rule block in both years. The model's output is a variable representing a forecast of the financial situation of an audited company. This variable ranges from 0 to 1, while it is

**for membership function** 

the threshold value for membership functions. These values are presented in Table 8.

**Ratio Symbol Threshold value** 

**X3\_1** 1.025 **X8\_1** 0.03 **X9\_1** 2.0 **X10\_1** (-0.1)

**X1\_2** 0.02 **X3\_2** 1.4 **X5\_2** 0.14 **X7\_2** 0.8 **X8\_2** 0.102

into the models as entry data nodes:

arbitrarily designated by the author.

information, i.e. financial ratios.

 one year prior bankruptcy – X3\_1, X8\_1, X9\_1, X10\_1, two years prior bankruptcy – X1\_2, X3\_2, X5\_2, X7\_2, X8\_2. assumed that the threshold value separating the "good" and "bad" companies is 0.5 (output variable values below 0,5 mean the company is at risk of bankruptcy, while those above 0.5 represent a company safe from bankruptcy). The final result generated by the fuzzy logic model is based on an assessment of four (one year analysis) and five financial ratios (two years analysis). The rule block in the model consists following set of rules for forecasting the economic situation in one year prior financial failure:

```
If X3_1 <= 1.025 and X8_1 <= 0.03 and X9_1 <= 2 and X10_1 <= (-0.1) then 0 
If X3_1 <= 1.025 and X8_1 <= 0.03 and X9_1 <= 2 and X10_1 > (-0.1) then 0 
If X3_1 <=1.025 and X8_1 <= 0.03 and X9_1 > 2 and X10_1 > (-0.1) then 0 
If X3_1 <=1.025 and X8_1 > 0.03 and X9_1 > 2 and X10_1 > (-0.1) then 1 
If X3_1 > 1.025 and X8_1 > 0.03 and X9_1 > 2 and X10_1 > (-0.1) then 1 
If X3_1 > 1.025 and X8_1 > 0.03 and X9_1 > 2 and X10_1 <= (-0.1) then 1 
If X3_1 > 1.025 and X8_1 > 0.03 and X9_1 <= 2 and X10_1 <= (-0.1) then 1 
If X3_1 > 1.025 and X8_1 <= 0.03 and X9_1 <= 2 and X10_1 <= (-0.1) then 0 
If X3_1 <=1.025 and X8_1 > 0.03 and X9_1 <= 2 and X10_1 <= (-0.1) then 0 
If X3_1 <=1.025 and X8_1 <= 0.03 and X9_1 > 2 and X10_1 <= (-0.1) then 0 
If X3_1 <=1.025 and X8_1 > 0.03 and X9_1 > 2 and X10_1 <= (-0.1) then 0 
If X3_1 > 1.025 and X8_1 <= 0.03 and X9_1 > 2 and X10_1 > (-0.1) then 1 
If X3_1 > 1.025 and X8_1 > 0.03 and X9_1 <= 2 and X10_1 > (-0.1) then 1 
If X3_1 > 1.025 and X8_1 <= 0.03 and X9_1 <= 2 and X10_1 > (-0.1) then 0 
If X3_1 > 1.025 and X8_1 <= 0.03 and X9_1 > 2 and X10_1 <= (-0.1) then 0 
If X3_1 <=1.025 and X8_1 > 0.03 and X9_1 <= 2 and X10_1 > (-0.1) then 0
```
A set of rules for forecasting the economic situation of companies in two years prior to bankruptcy is as follows:

If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 > 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1 If X1\_2 > 0.02 and X5\_2 > 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 <= 1.4 and X7\_2 > 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 > 0.8 then 0 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 > 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 1 If X1\_2 <= 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 <= 0.02 and X5\_2 > 0.14 and X8\_2 > 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 0 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1 If X1\_2 > 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1 If X1\_2 > 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 <= 1.4 and X7\_2 > 0.8 then 1 If X1\_2 > 0.02 and X5\_2 > 0.14 and X8\_2 > 0.102 and X3\_2 <= 1.4 and X7\_2 <= 0.8 then 1 If X1\_2 > 0.02 and X5\_2 > 0.14 and X8\_2 <= 0.102 and X3\_2 > 1.4 and X7\_2 <= 0.8 then 1 If X1\_2 > 0.02 and X5\_2 <= 0.14 and X8\_2 > 0.102 and X3\_2 > 1.4 and X7\_2 > 0.8 then 1

Fuzzy Logic in Financial Management 281

After creating the two fuzzy logic models the author, using the same financial ratios, programmed two artificial neural networks based on the same learning dataset. The aim of such a research approach is to compare the effectiveness of an innovative forecasting method in economics – fuzzy logic (until 2006, the use of fuzzy logic in finance and economics was practically unknown4), with the most popular method of soft computing techniques. By using the same population of enterprises to develop models, author is able to

 one year prior to bankruptcy – 4 input neurons (financial ratios: X3\_1, X8\_1, X9\_1, X10\_1), 9 hidden neurons where mathematical calculations were made, and 2 output

 two years prior to bankruptcy – 5 input neurons (financial ratios: X1\_2, X3\_2, X5\_2, X7\_2, X8\_2), 10 hidden neurons, 2 output neurons (0 – bankrupt "BR", 1 – non-bankrupt

In the last stage of this research, the author analyzed the efficiency of the discriminant analysis model created by Altman in forecasting business bankruptcy one year and two years before, based on testing dataset "one" and "two". The form of this model can be found in Section 2 of this chapter. The aim of such comparison is to analyze the usefulness of the first bankruptcy model created in 1968 (which is still the most popular and widely used in the business world) on the same population of companies as in case of developed fuzzy

4 Author of this chapter has not found any papers on the use of fuzzy logic in forecasting bankruptcy of

Fig. 10. Membership Functions for Variable "X8\_2".

verify their effectiveness and to identify the most effective model.

neurons (0 – bankrupt "BR", 1 – non-bankrupt "NBR"),

"NBR") – Figure 11.

entities before 2006.

logic and artificial neural network models.

The architecture of developed models by author of this chapter is as follows:

Fig. 8. The Complete Block Diagram of the Fuzzy Logic Model for Business Credit Scoring

The exemplary form of the membership functions are presented in Figure 9 for the variable "X3\_2" and in Figure 10 for variable "X8\_2".

Fig. 9. Membership Functions for Variable "X3\_2".

Fig. 8. The Complete Block Diagram of the Fuzzy Logic Model for Business Credit Scoring

The exemplary form of the membership functions are presented in Figure 9 for the variable

**R U L E**

**B L O C K**

**ENTRY VARIABLES** 

"X3\_2" and in Figure 10 for variable "X8\_2".

Fig. 9. Membership Functions for Variable "X3\_2".

Fig. 10. Membership Functions for Variable "X8\_2".

After creating the two fuzzy logic models the author, using the same financial ratios, programmed two artificial neural networks based on the same learning dataset. The aim of such a research approach is to compare the effectiveness of an innovative forecasting method in economics – fuzzy logic (until 2006, the use of fuzzy logic in finance and economics was practically unknown4), with the most popular method of soft computing techniques. By using the same population of enterprises to develop models, author is able to verify their effectiveness and to identify the most effective model.

The architecture of developed models by author of this chapter is as follows:


In the last stage of this research, the author analyzed the efficiency of the discriminant analysis model created by Altman in forecasting business bankruptcy one year and two years before, based on testing dataset "one" and "two". The form of this model can be found in Section 2 of this chapter. The aim of such comparison is to analyze the usefulness of the first bankruptcy model created in 1968 (which is still the most popular and widely used in the business world) on the same population of companies as in case of developed fuzzy logic and artificial neural network models.

<sup>4</sup> Author of this chapter has not found any papers on the use of fuzzy logic in forecasting bankruptcy of entities before 2006.

Fuzzy Logic in Financial Management 283

models. The effectiveness of the fuzzy logic model decreased by 3.7 percentage points (from 87.03% one year before bankruptcy to 83.33% two years prior to insolvency). In the case of the artificial neural network effectiveness decreased by 18.52 percentage points (from 87.03% to 68.51%) and in the case of the discriminant analysis model prediction quality

Table 9. The Results of Effectiveness of Fuzzy Logic Model (FL), Artificial Neural Network (ANN) and Discriminant Analysis Model (DA) in Forecasting Business Bankruptcies

Similarly at it was in case of consumer credit scoring, due to the equal distribution of bankrupt and non-bankrupt companies in the testing dataset "one", the author treats this research approach as a theoretical possibility test of the predictive power of methods used. From the viewpoint of the practical applicability of these methods in business, the conclusions from the tests conducted on testing dataset "two", which contained 81% companies with good financial condition and less than 19% firms at risk of insolvency, are more important to analyze. Figure 12 shows that in such circumstances, the fuzzy logic

 in the analysis one year prior to bankruptcy S1: by 8.34 percentage points better than artificial neural network and by 23.49 percentage points better than the discriminant

 in the analysis two years prior to bankruptcy S2: by 3.03 percentage points better than artificial neural network model and by as much as 15.90 percentage points better than

It is also worth mentioning that despite a small difference of overall effectiveness S2 in analysis of two years prior to bankruptcy between fuzzy logic model and artificial neural networks (65.90% vs. 62.87%) – figure 12, the artificial neural networks generated six times greater I type errors E1 than fuzzy logic model (24% vs. 4%), and discriminant analysis model made seven times greater errors of such type than fuzzy logic model (28% vs. 4%) – see figure 13. As it was explained before – such errors are much more costly to make by banks than II type errors. Figure 13 shows also that discriminant analysis model additionally generated much greater II

Evaluating I type (E1) and II type (E2) errors in one year prior financial failure of enterprises, it can be said that discriminant analysis model made only 8 percentage points greater I type errors than both fuzzy logic and artificial neural network model (24% vs. 16%) – figure 14. But there is a huge difference in II type errors between analyzed models. Figure 14 shows that DA model generated as much as 46.72% of II type errors, while fuzzy logic model only 19.62%.

the discriminant analysis model (effectiveness 65.90% vs. 62.87% and vs. 50%).

analysis model (effectiveness: 81.06% vs. 72.72% and vs. 57.57%);

type errors E2 than fuzzy logic and artificial neural networks.

**DA (Altman)**  **Method** 

E1 24% (6) 16% (4) 16% (4) E2 20.68 (6) 10.34% (3) 10.34% (3) **S 77.77% 87.03% 87.03%** 

E1 28% (7) 24% (6) 4% (1) E2 41.37% (12) 37.93% (11) 27.58% (8) **S 64.81% 68.51% 83.33%** 

**ANN (Korol)** 

**FL (Korol)** 

decreased by 12.96 percentage points (from 77.77% to 64.81%).

**Testing Type Time Effectiveness** 

**One year before** 

**Two years before** 

model achieved greater overall effectiveness:

(parentheses contain the number of misclassified firms).

Testing dataset "one" 25:29

Fig. 11. Architecture of the Artificial Neural Network Model for Evaluating Polish Enterprises in the Analysis of Two Years Prior To Bankruptcy

#### **4.3 The results**

The results obtained from testing the two fuzzy logic models and two artificial neural networks developed against the bankruptcy risk of enterprises while testing dataset "one" and "two" are presented in Table 9.

The tests carried out on dataset "one" showed in the analysis one year prior to bankruptcy that the fuzzy logic model obtained 87.03% effectiveness. The same effectiveness was generated by the artificial neural networks model. Table 9 shows, however, that as the forecasting period increases to two years before bankruptcy, the fuzzy logic model is characterized by much better predictive properties than the artificial neural networks model (83.33% vs. 68.51%). In both years of analysis, the discriminant analysis model was worse than the fuzzy logic model (by 9.26 percentage points – one year before, and by as much as 18.52 percentage points – two years before) and artificial neural networks (by 9.26 percentage points – one year before, and by 3.7 percentage points – two years prior to bankruptcy). It is also necessary to point out that in the fuzzy logic model case the decrease of effectiveness with increased period of forecast is the smallest compared to the other two

1

2

3

4

5

6

7

8

9

10

NBR

BR

Fig. 11. Architecture of the Artificial Neural Network Model for Evaluating Polish

The results obtained from testing the two fuzzy logic models and two artificial neural networks developed against the bankruptcy risk of enterprises while testing dataset "one"

The tests carried out on dataset "one" showed in the analysis one year prior to bankruptcy that the fuzzy logic model obtained 87.03% effectiveness. The same effectiveness was generated by the artificial neural networks model. Table 9 shows, however, that as the forecasting period increases to two years before bankruptcy, the fuzzy logic model is characterized by much better predictive properties than the artificial neural networks model (83.33% vs. 68.51%). In both years of analysis, the discriminant analysis model was worse than the fuzzy logic model (by 9.26 percentage points – one year before, and by as much as 18.52 percentage points – two years before) and artificial neural networks (by 9.26 percentage points – one year before, and by 3.7 percentage points – two years prior to bankruptcy). It is also necessary to point out that in the fuzzy logic model case the decrease of effectiveness with increased period of forecast is the smallest compared to the other two

Enterprises in the Analysis of Two Years Prior To Bankruptcy

**4.3 The results** 

X8\_2

X7\_2

X5\_2

X3\_2

X1\_2

and "two" are presented in Table 9.

models. The effectiveness of the fuzzy logic model decreased by 3.7 percentage points (from 87.03% one year before bankruptcy to 83.33% two years prior to insolvency). In the case of the artificial neural network effectiveness decreased by 18.52 percentage points (from 87.03% to 68.51%) and in the case of the discriminant analysis model prediction quality decreased by 12.96 percentage points (from 77.77% to 64.81%).


Table 9. The Results of Effectiveness of Fuzzy Logic Model (FL), Artificial Neural Network (ANN) and Discriminant Analysis Model (DA) in Forecasting Business Bankruptcies (parentheses contain the number of misclassified firms).

Similarly at it was in case of consumer credit scoring, due to the equal distribution of bankrupt and non-bankrupt companies in the testing dataset "one", the author treats this research approach as a theoretical possibility test of the predictive power of methods used. From the viewpoint of the practical applicability of these methods in business, the conclusions from the tests conducted on testing dataset "two", which contained 81% companies with good financial condition and less than 19% firms at risk of insolvency, are more important to analyze. Figure 12 shows that in such circumstances, the fuzzy logic model achieved greater overall effectiveness:


It is also worth mentioning that despite a small difference of overall effectiveness S2 in analysis of two years prior to bankruptcy between fuzzy logic model and artificial neural networks (65.90% vs. 62.87%) – figure 12, the artificial neural networks generated six times greater I type errors E1 than fuzzy logic model (24% vs. 4%), and discriminant analysis model made seven times greater errors of such type than fuzzy logic model (28% vs. 4%) – see figure 13. As it was explained before – such errors are much more costly to make by banks than II type errors. Figure 13 shows also that discriminant analysis model additionally generated much greater II type errors E2 than fuzzy logic and artificial neural networks.

Evaluating I type (E1) and II type (E2) errors in one year prior financial failure of enterprises, it can be said that discriminant analysis model made only 8 percentage points greater I type errors than both fuzzy logic and artificial neural network model (24% vs. 16%) – figure 14. But there is a huge difference in II type errors between analyzed models. Figure 14 shows that DA model generated as much as 46.72% of II type errors, while fuzzy logic model only 19.62%.

Fuzzy Logic in Financial Management 285

The research conducted showed that it is worth developing such early warning models. All presented fuzzy logic models in the chapter are characterized by high forecasting effectiveness. The author has proven that fuzzy logic can be a very useful and powerful tool in financial analysis, even though the use of fuzzy logic in finance was practically unknown until 2006. Therefore, it is one of the first attempts at using fuzzy logic to predict enterprise and consumer bankruptcy in worldwide literature. The developed bankruptcy prediction models presented in this chapter can be easily used by financial managers as a decisional aid

tool in the process of evaluating the financial situation of enterprises and consumers.

transforming the set of decision rules for individual needs.

achieved, but also in terms of three aspects:

need to re-estimate the entire model.


companies and consumers.

pp. 285-300

**6. References** 

It should be emphasized that the fuzzy logic models presented have high practical values. Due to the fact that these models are an "open" application, a person interested in its use can not only use them in their current form, but can also easily modify them for their own needs. For example, a person managing an international company can add exchange rate as a risk factor to the model. The number of model adaptations is virtually unlimited by

The models presented are superior to even the sophisticated methods of artificial intelligence, such as artificial neural network models, not only in terms of effectiveness



It is necessary to note that the aim of this paper was to evaluate the efficiency of fuzzy logic model in forecasting the financial situation of companies and households and to give the reader "the opened" structure of fuzzy logic model, that can be easily adopted to changed economic situation in the country or even adopted for implementation in different country or region of the world. Therefore, despite the fact that the presented research (both consumer and business credit scoring) is based on financial data from the years 2000 – 2007, it is still valid (the value of variables used did not change significantly in the economy) and useful tool to use nowadays and in future with adopting individual variables (for example – the monthly income of customer etc.). To summarize, this chapter provides the reader with practical models that can be used in financial management. Such models are an useful tool that can be both updated with the passage of time, and adopted for individual needs.

The conclusions of these studies can also be applied to other European, American or Asian

Agarwal, V. & Taffler, R. (2007). Twenty-five years of the Taffler z-score model – does it

really have predictive ability? *Accounting and Business Research*, Vol. 37, No 4, 2007,

artificial neural network, which operates on the "black box" principle),

**5. Conclusions** 

Fig. 12. The Results of Overall Effectiveness of Fuzzy Logic Model (FL), Artificial Neural Network (ANN) and Discriminant Analysis Model (DA) in Forecasting Business Bankruptcies – Testing Dataset "Two" – 25:107.

Fig. 13. The Results of Generated I and II Type Errors by Fuzzy Logic Model (FL), Artificial Neural Network (ANN) and Discriminant Analysis Model (DA) in Forecasting Business Bankruptcies – Testing Dataset "Two" – 25:107 – Two Years Prior Bankruptcy.

Fig. 14. The Results of Generated I and II Type Errors by Fuzzy Logic Model (FL), Artificial Neural Network (ANN) and Discriminant Analysis Model (DA) in Forecasting Business Bankruptcies – Testing Dataset "Two" – 25:107 – One Year Prior Bankruptcy.

The above conclusions regarding the overall effectiveness, I type and II type errors proved the superiority of developed fuzzy logic models for both years of analyses over the model of discriminant analysis and artificial neural networks.

#### **5. Conclusions**

284 Fuzzy Logic – Emerging Technologies and Applications

Fig. 12. The Results of Overall Effectiveness of Fuzzy Logic Model (FL), Artificial Neural

Fig. 13. The Results of Generated I and II Type Errors by Fuzzy Logic Model (FL), Artificial Neural Network (ANN) and Discriminant Analysis Model (DA) in Forecasting Business

Fig. 14. The Results of Generated I and II Type Errors by Fuzzy Logic Model (FL), Artificial Neural Network (ANN) and Discriminant Analysis Model (DA) in Forecasting Business

The above conclusions regarding the overall effectiveness, I type and II type errors proved the superiority of developed fuzzy logic models for both years of analyses over the model of

Bankruptcies – Testing Dataset "Two" – 25:107 – One Year Prior Bankruptcy.

discriminant analysis and artificial neural networks.

Bankruptcies – Testing Dataset "Two" – 25:107 – Two Years Prior Bankruptcy.

Network (ANN) and Discriminant Analysis Model (DA) in Forecasting Business

Bankruptcies – Testing Dataset "Two" – 25:107.

The research conducted showed that it is worth developing such early warning models. All presented fuzzy logic models in the chapter are characterized by high forecasting effectiveness. The author has proven that fuzzy logic can be a very useful and powerful tool in financial analysis, even though the use of fuzzy logic in finance was practically unknown until 2006. Therefore, it is one of the first attempts at using fuzzy logic to predict enterprise and consumer bankruptcy in worldwide literature. The developed bankruptcy prediction models presented in this chapter can be easily used by financial managers as a decisional aid tool in the process of evaluating the financial situation of enterprises and consumers.

It should be emphasized that the fuzzy logic models presented have high practical values. Due to the fact that these models are an "open" application, a person interested in its use can not only use them in their current form, but can also easily modify them for their own needs. For example, a person managing an international company can add exchange rate as a risk factor to the model. The number of model adaptations is virtually unlimited by transforming the set of decision rules for individual needs.

The models presented are superior to even the sophisticated methods of artificial intelligence, such as artificial neural network models, not only in terms of effectiveness achieved, but also in terms of three aspects:


It is necessary to note that the aim of this paper was to evaluate the efficiency of fuzzy logic model in forecasting the financial situation of companies and households and to give the reader "the opened" structure of fuzzy logic model, that can be easily adopted to changed economic situation in the country or even adopted for implementation in different country or region of the world. Therefore, despite the fact that the presented research (both consumer and business credit scoring) is based on financial data from the years 2000 – 2007, it is still valid (the value of variables used did not change significantly in the economy) and useful tool to use nowadays and in future with adopting individual variables (for example – the monthly income of customer etc.). To summarize, this chapter provides the reader with practical models that can be used in financial management. Such models are an useful tool that can be both updated with the passage of time, and adopted for individual needs.

The conclusions of these studies can also be applied to other European, American or Asian companies and consumers.

#### **6. References**

Agarwal, V. & Taffler, R. (2007). Twenty-five years of the Taffler z-score model – does it really have predictive ability? *Accounting and Business Research*, Vol. 37, No 4, 2007, pp. 285-300

**14** 

*México* 

**Fuzzy Modeling of Geospatial Patterns** 

In computer science, the design of intelligent systems able to manage uncertain, indefinite or incomplete information is called Soft Computing (Zadeh 1994). Its aim is to illustrate real problems that are not manageable by conventional techniques. The main techniques that compose Soft Computing are fuzzy logic, neural networks, evolutionary computing and probabilistics. The works published by Prof. Zadeh on fuzzy sets, fuzzy logic, fuzzy systems, neural networks, soft computing and computing with words have had applications in a great diversity of areas—computational modeling, optimizing, planning, control,

Geospatial modeling and retrieving geographical information has become an important part of different areas of knowledge, such as environmental science, urban planning and criminal spatial patterns, among others. This work examines some of the fuzzy tools most commonly used in geospatial modeling for spatial analysis and image processing. We will present a family of models as an alternative to the fuzzy mathematical representation of spatial patterns. This chapter is primarily concerned with spatial pattern methodologies that attempt to describe the arrangement of phenomena in space. In most cases, these phenomena have either point or area features. Point and area analyses use randomness (or a lack of pattern) as a dividing point between two opposite pattern types—dispersed and clustered. This work also presents a general framework (fuzzy data fusion) to combine information from several individual classifications obtained from satellite images in order to

The principal ideas and concepts of fuzzy logic (FL), as shown by Zadeh, are that FL is a precise logic of uncertainty and approximate reasoning (Zadeh 1975, 1976). Zadeh (2010) notes two ideas that FL takes from human capabilities. The first refers to an environment of imperfect information that includes uncertainty, incompleteness of information, conflicting information, partiality of truth and partiality of possibility. The second relates to the capability to perform a wide variety of physical and mental tasks without any

geospatial analysis, image classification, prediction and image fusion.

recognize spatial patterns and improve spatial pattern extraction.

**1. Introduction** 

**2. General framework** 

measurements or computations.

Alejandra A. López-Caloca and Carmen Reyes *Centro de Investigación en Geografía y Geomática* 

*"Jorge L. Tamayo" A.C., CentroGeo,* 


## **Fuzzy Modeling of Geospatial Patterns**

Alejandra A. López-Caloca and Carmen Reyes

*Centro de Investigación en Geografía y Geomática "Jorge L. Tamayo" A.C., CentroGeo, México* 

#### **1. Introduction**

286 Fuzzy Logic – Emerging Technologies and Applications

Altman, E. & Rijken, H. (2006). A point-in-time perspective on through-the-cycle ratings,

Aziz, M. & Dar, H. (2001). Predicting corporate bankruptcy – where we stand? *Corporate* 

Bose, I. & Mahapatra, R. (2001). Business data mining – a machine learning perspective,

Boyle, M.; Crook, J.; Hamilton, R. & Thomas L. (1992). Methods for credit scoring applied to

Henley, W. & Hand, D. (1996). A k-NN classifier for assessing consumer credit risk, *The* 

Korol, T. (2011). Multi-Criteria Early Warning System Against Enterprise Bankruptcy Risk, *International Research Journal of Finance and Economics*, issue 61, pp. 141-154 Kumar, P. & Ravi, V. (2007). Bankruptcy prediction in banks and firms via statistical and

Mcleay, S. & Omar, A. (2000). The sensitivity of prediction models to the non-normality of

Nwogugu, M. (2007). Decision-making, risk and corporate governance – a critique of

Ooghe, H. & Balcaen, S. (2006). 35 years of studies on business failure – an overview of the

Thomas, L. (2000). Survey of credit and behavioural scoring – forecasting financial risk of lending to consumers, *International Journal of Forecasting*, No. 16, pp. 149-172 Tingting, J. (2006). Consumer credit delinquency and bankruptcy forecasting using

Wiginton, J. (1980). A note on the comparison of logit and discriminant models of consumer credit behaviour, *Journal of Financial and Quantitative Analysis*, No. 15, pp. 757-770 Wilson, L. & Sharda, R. (1994). Bankruptcy prediction using neural networks, *Decision* 

intelligent techniques – a review, *European Journal of Operational Research*, No. 180,

bounded and unbounded financial ratios, *British Accounting Review*, No. 32, pp.

methodological issues in bankruptcy/recovery prediction models, *Applied* 

classic statistical methodologies and their related problems, *The British Accounting* 

*Financial Analysts Journal*, No. 62/1, pp. 54-70

*Governance Journal*, Vol. 6, No. 1, pp. 18-33

*Statistician*, No. 65, pp. 77-95

*Review*, No. 38, pp. 63-93

*Support Systems*, No. 11, pp. 548-550

pp. 1-28

213-230

Altman, E. (1993). Corporate financial distress, *John Wiley & Sons*, New York

*Information and Management Journal*, No. 39, pp. 211-225

slow payers, *Oxford University Press*, Oxford, pp. 75-90

*Mathematics and Computation*, No. 185, pp. 178-196

advanced econometric modeling, *MPRA Paper*, No. 3187

Zadeh, L. (1965). Fuzzy sets, *Information and Control*, No. 8 (3), pp. 338-353.

In computer science, the design of intelligent systems able to manage uncertain, indefinite or incomplete information is called Soft Computing (Zadeh 1994). Its aim is to illustrate real problems that are not manageable by conventional techniques. The main techniques that compose Soft Computing are fuzzy logic, neural networks, evolutionary computing and probabilistics. The works published by Prof. Zadeh on fuzzy sets, fuzzy logic, fuzzy systems, neural networks, soft computing and computing with words have had applications in a great diversity of areas—computational modeling, optimizing, planning, control, geospatial analysis, image classification, prediction and image fusion.

Geospatial modeling and retrieving geographical information has become an important part of different areas of knowledge, such as environmental science, urban planning and criminal spatial patterns, among others. This work examines some of the fuzzy tools most commonly used in geospatial modeling for spatial analysis and image processing. We will present a family of models as an alternative to the fuzzy mathematical representation of spatial patterns. This chapter is primarily concerned with spatial pattern methodologies that attempt to describe the arrangement of phenomena in space. In most cases, these phenomena have either point or area features. Point and area analyses use randomness (or a lack of pattern) as a dividing point between two opposite pattern types—dispersed and clustered. This work also presents a general framework (fuzzy data fusion) to combine information from several individual classifications obtained from satellite images in order to recognize spatial patterns and improve spatial pattern extraction.

#### **2. General framework**

The principal ideas and concepts of fuzzy logic (FL), as shown by Zadeh, are that FL is a precise logic of uncertainty and approximate reasoning (Zadeh 1975, 1976). Zadeh (2010) notes two ideas that FL takes from human capabilities. The first refers to an environment of imperfect information that includes uncertainty, incompleteness of information, conflicting information, partiality of truth and partiality of possibility. The second relates to the capability to perform a wide variety of physical and mental tasks without any measurements or computations.

Fuzzy Modeling of Geospatial Patterns 289

Because dynamic geospatial processes and transformations occurring over long periods of time are not uniform, it is necessary to consider changes in their spatial attributes with their temporal dimensional. For example, geometric spatial changes are investigated with what is known as fuzzy change detection (Molenaar & Cheng , 2000). Different classes of spatial changes in frontier areas must be considered with respect to diverse observations at different times (changes in dimension, connectivity, size, shape and non-geometrical spatial attributes). These are often difficult to determine because of a lack of dimensions at the time

Objects are defined according to a geospatial scale and context. The observation is related to the scale at which the object is described (Couclelis, 1996). When having satellite images with different levels of spatial resolution, the identification of more classes will increase by having more detail on these images. The degree of uncertainty of many geographic objects, with respect to the scale of observation, is proposed on the basis of a neural network structure approach (Silván-Cardenas, 2008) and particular data representation models of objects with scale-induced indeterminate boundaries. Using this approach, fuzzy points and fuzzy lines are considered and the connection between the degree of fuzziness and the scale

Spatial data are important to diverse studies; in fact, new technology continually enables generating new data. A number of available methods are applied to spatial data, some of which include spatial classification. Nevertheless, difficulties exist that can be conceptualized and modeled with fuzzy concepts, for example, by eliminating strict ideas of

In geo-modeling, the utilization of fuzzy concepts with uncertainty problems (Cheng, 2002,

a. Spatial Incompleteness. Indetermination is related to objects that cannot be separated, or where there is a lack of information (incompleteness) and imprecision. This can be due to the particular information not covering a specific region or the definition of categories that only makes sense in certain parts of the space (e.g meteorological measurements); in remote sensing, for example, the presence of clouds and shadows on

b. Fuzziness. To construct better real-world models, it is necessary to understand the concept of fuzziness as unsharpness of class boundaries, as well as the role of precision in fuzzy borders. In regionalization studies, regions are defined so that every element in the study universe is distinguishable regardless of whether or not it belongs to the region. Geographical problems can undoubtedly benefit from the fuzzy definition. For example, in the study of urban areas, a characteristic of the city is its lack of clear-cut differences in residential areas as well as in land use, for which incorporating the

definition of fuzzy regions is therefore more appropriate (Reyes-Guerrero, 1986). c. Time Incoherence. Temporal uncertainty with respect to time is common when the phenomena observed occurs when precise knowledge is not available about the instant of such information, and it only can be approximated with certain measurements. The value of the information depends on time, since many observed phenomena have

they occur.

of representation is discussed.

encountering boundaries on the geographic objects studied.

satellite images that do not obstruct observation of the zone.

temporal space relations, such as vegetation-season.

Cheng et al 2004) can be divided in four ways:

For example, when considering fuzzy concepts, we talk about the lack of sharp class boundaries. Thus, the starting point for FL is the fuzzy set concept, where a fuzzy set is a class with unsharp boundaries. FL deals with three basic concepts: graduation, granulation, and precisiation (Zadeh, 2010). The graduation concept is associated with scale- and membership-degree functions. The granulation concept is useful with regard to imprecision, uncertainty and complexity. For precisiation, two approximations are defined—precisiation and imprecisiation, where the former is based on measurements and the latter on perceptions. In fact, the precisiation of meaning has always played an important role in science. Therefore, graduation is related to precisiation and granulation to imprecisiation.

#### **2.1 Geospatial modeling**

The modeling of natural phenomena requires knowledge of the geographic landscape entities that can be conceptualized in space (places, axiomatic geometry, point-set topology, discrete space, raster/vector), spatial attributes and relations (dimension, connectivity, position, size, location, shape), thematic (natural and conventional objects, classifying objects, pattern recognition) and temporal forms (states, processes and dynamic events). The different fields—geography, biology, hydrology, geology, remote sensing, ecology, and others—select the most important aspects of a phenomenon, i.e., representative variables, to generate data (Jacquez et al., 2000) and perform modeling.

Natural objects that are characterized and define variables may be highly regular, in which a large number of cases are shown as an individual, easy-to-identify elements. Others, however, tend to be highly irregular, fragmented, fuzzier, and have boundaries that are difficult to describe (Galton, 2000). A natural pattern is generated by various processes at different space and time scales depending on the phenomena being investigated; hence the interest in fuzzy modeling for research to illustrate geographic problems (Altman, 1994; Usery 1996; Molenaar & Cheng, 2000, Croos & Firat, 2004, Guesgen, 2005).

Problems involving indeterminate boundaries—continuity, heterogeneity, dynamics, scaledependence (Cheng, 2002; Burroughs, 1996) and contiguity—found in the very nature of objects are described below.

Continuity refers to continuous space, which is seamless, i.e., two regions are not separated but rather are distributed in a continuous way in space. In some cases, their distribution does not permit identifying a very precise border because neither the objects nor isolated processes exist. The problem is to represent these objects in a discrete space (Kavoras, 1996). The nature of the object influences how we become aware of the boundaries and their degree of sharpness (Erwig & Schneider, 1997).

Contiguity measurements evaluate the characteristics of spatial features that are connected, i.e., the evaluation of features that touch one another, that are near one another. Contiguity is desirable to reduce the negative environmental impact on forests, where forest patches affect interior forest habitats.

Spatial heterogeneity is the existence of each object or entity in relation to others, as well as the attributes and qualities of each one of them, and is determined at the moment of mutual interaction. This explains two or various types of vegetation existing in a forest zone and, therefore, describes a heterogeneity problem. The similarity or difference between an entity and its surrounding is a measure of this variation (Reyes-Guerrero, 1986).

For example, when considering fuzzy concepts, we talk about the lack of sharp class boundaries. Thus, the starting point for FL is the fuzzy set concept, where a fuzzy set is a class with unsharp boundaries. FL deals with three basic concepts: graduation, granulation, and precisiation (Zadeh, 2010). The graduation concept is associated with scale- and membership-degree functions. The granulation concept is useful with regard to imprecision, uncertainty and complexity. For precisiation, two approximations are defined—precisiation and imprecisiation, where the former is based on measurements and the latter on perceptions. In fact, the precisiation of meaning has always played an important role in science. Therefore, graduation is related to precisiation and granulation to imprecisiation.

The modeling of natural phenomena requires knowledge of the geographic landscape entities that can be conceptualized in space (places, axiomatic geometry, point-set topology, discrete space, raster/vector), spatial attributes and relations (dimension, connectivity, position, size, location, shape), thematic (natural and conventional objects, classifying objects, pattern recognition) and temporal forms (states, processes and dynamic events). The different fields—geography, biology, hydrology, geology, remote sensing, ecology, and others—select the most important aspects of a phenomenon, i.e., representative variables, to

Natural objects that are characterized and define variables may be highly regular, in which a large number of cases are shown as an individual, easy-to-identify elements. Others, however, tend to be highly irregular, fragmented, fuzzier, and have boundaries that are difficult to describe (Galton, 2000). A natural pattern is generated by various processes at different space and time scales depending on the phenomena being investigated; hence the interest in fuzzy modeling for research to illustrate geographic problems (Altman, 1994;

Problems involving indeterminate boundaries—continuity, heterogeneity, dynamics, scaledependence (Cheng, 2002; Burroughs, 1996) and contiguity—found in the very nature of

Continuity refers to continuous space, which is seamless, i.e., two regions are not separated but rather are distributed in a continuous way in space. In some cases, their distribution does not permit identifying a very precise border because neither the objects nor isolated processes exist. The problem is to represent these objects in a discrete space (Kavoras, 1996). The nature of the object influences how we become aware of the boundaries and their

Contiguity measurements evaluate the characteristics of spatial features that are connected, i.e., the evaluation of features that touch one another, that are near one another. Contiguity is desirable to reduce the negative environmental impact on forests, where forest patches

Spatial heterogeneity is the existence of each object or entity in relation to others, as well as the attributes and qualities of each one of them, and is determined at the moment of mutual interaction. This explains two or various types of vegetation existing in a forest zone and, therefore, describes a heterogeneity problem. The similarity or difference between an entity

**2.1 Geospatial modeling** 

objects are described below.

affect interior forest habitats.

degree of sharpness (Erwig & Schneider, 1997).

generate data (Jacquez et al., 2000) and perform modeling.

Usery 1996; Molenaar & Cheng, 2000, Croos & Firat, 2004, Guesgen, 2005).

and its surrounding is a measure of this variation (Reyes-Guerrero, 1986).

Because dynamic geospatial processes and transformations occurring over long periods of time are not uniform, it is necessary to consider changes in their spatial attributes with their temporal dimensional. For example, geometric spatial changes are investigated with what is known as fuzzy change detection (Molenaar & Cheng , 2000). Different classes of spatial changes in frontier areas must be considered with respect to diverse observations at different times (changes in dimension, connectivity, size, shape and non-geometrical spatial attributes). These are often difficult to determine because of a lack of dimensions at the time they occur.

Objects are defined according to a geospatial scale and context. The observation is related to the scale at which the object is described (Couclelis, 1996). When having satellite images with different levels of spatial resolution, the identification of more classes will increase by having more detail on these images. The degree of uncertainty of many geographic objects, with respect to the scale of observation, is proposed on the basis of a neural network structure approach (Silván-Cardenas, 2008) and particular data representation models of objects with scale-induced indeterminate boundaries. Using this approach, fuzzy points and fuzzy lines are considered and the connection between the degree of fuzziness and the scale of representation is discussed.

Spatial data are important to diverse studies; in fact, new technology continually enables generating new data. A number of available methods are applied to spatial data, some of which include spatial classification. Nevertheless, difficulties exist that can be conceptualized and modeled with fuzzy concepts, for example, by eliminating strict ideas of encountering boundaries on the geographic objects studied.

In geo-modeling, the utilization of fuzzy concepts with uncertainty problems (Cheng, 2002, Cheng et al 2004) can be divided in four ways:


Fuzzy Modeling of Geospatial Patterns 291

The FCM algorithm utilized for this task is based on the minimization of the objective function (eq. 1), which represents the distance from any given data point to a cluster center, weighted by the membership grade of that data point. By iteration, the cluster centers and the membership grade are updated, and the objective function is minimized to find the best

1 1

ǁ ǁ is any norm expressing the similarity between any measured data and the center*,* 

C is the number of clusters, the parameter *m* is the weighted exponent for *uij* and controls

*m* is any real number greater than 1 and is called the fuzzifier parameter, for which 2 is

Fuzzy partitioning is carried out through an iterative optimization of the objective function

2 1

*xc xc*

*<sup>N</sup> <sup>m</sup> ij i*

1

*ij i*

*u*

*u x*

1

/ *ij <sup>C</sup> <sup>m</sup> ij iu*

> *i <sup>j</sup> <sup>N</sup> <sup>m</sup>*

 

a. Initialize U (membership matrix), called the fuzzy partition matrix, where *uij* denotes

<sup>0</sup> , *U u matrix U ij* ;

*<sup>k</sup> C cj* with *k <sup>U</sup>* ;

where <sup>1</sup>

*c*

*xi* is the *i-*th of d-dimensional measured data, *cj* is the centroid of the cluster,

shown above, with the updating of membership *uij* and the cluster centers *cj* by:

1

*u*

*u*

The algorithm is comprised of the following steps:

b. Compute the vectors of the center prototypes

c. Compute the distances: <sup>2</sup> *DijA*

the membership degree of a datum xi to cluster *i*,

dij =ǁ xi-cjǁ2, is the distances of the pattern xi to the cluster centroid cj,

*uij* is the degree of membership of *xi* in the cluster *j*,

*C N <sup>m</sup> m ij i j j i J ux c* 

2

,

(1)

(2)

location of the clusters.

N is the number of data points,

usually chosen.

the "fuzziness" of the resulting cluster,

where

d. Other general problems to be considered regarding spatial data are the lack of data or of definition of the object studied, inexact data, inconsistent data from multiple sources, data processing errors, inadequate generalization operations, limitations in the spatial representation scheme and limitations in data acquisition technology (Burroughs, 1996).

### **3. Fuzzy spatial clustering**

In general, cluster analysis involves a set of data in groups or clusters that is organized so that items in the same group are similar to each other and different from those in other groups. In spatial information in clustering, different types of clustering analyses have been studied, including spatial clustering (clustering of spatial points), regionalization (clustering with geographic contiguity constraints) and point pattern analysis (hot-spot detection with spatial scan statistics). The use of many of these techniques for hot-spot detection is relatively problematic for several reasons, including the relatively arbitrary definition of the number of clusters to be included and the procedures applied to draw hot-spot boundaries. These contour areas indicate high to low robbery occurrence and, therefore, respond to the demand for public safety or provide alternatives to precisely locate schools in geographic distribution plans.

Fuzzy clustering methods allow objects to belong to several clusters simultaneously with different degrees of membership; in this work, we used Fuzzy C-Means clustering (FCM), (Bezdek, 1973). FCM is a data clustering technique that considers each data point belonging to a cluster to a certain degree, as specified by a membership degree. Two geospatial models with different applications are presented—hot-spot crime detection and educational planning.

#### **3.1 Spatial analysis of crime**

By definition, a hot-spot is a geographic area that presents a greater concentration of events as compared to its surroundings. It is an important tool for the analysis of point data to describe criminal activities, their geospatial distribution, and especially trends—in order to determine zones more likely to have higher concentrations of criminal events. The algorithms utilized to define a hot-spot may vary significantly when determining optimal and representative clusters—i.e., an adequate grouping must be determined. Generally, analysts must examine a series of possible solutions to spatially determine the optimal configuration; for example, cases using known methods such as hard clustering.

The strict assignment of parameters in the hard-clustering algorithm prevents identifying the optimal number of groups and, therefore, the result is not always realistic. Grubesic (2006) focuses on fuzzy grouping in the case of delinquency.

In order to disclose spatial crime patterns, Lopez-Caloca et al. (2009) tested criminal spatial patterns in the Mexican city of Hermosillo, as well as moving robberies (vehicle theft and public transportation robbery), fixed robberies (household or commercial establishment robberies) and violent robberies. Geo-referenced data from police records are available for each of the events reported during 2005 and 2006. The advantage of fuzzy clusters is a closer estimation of the delimitation of boundaries based on the information, with which to analyze processes in the region.

The FCM algorithm utilized for this task is based on the minimization of the objective function (eq. 1), which represents the distance from any given data point to a cluster center, weighted by the membership grade of that data point. By iteration, the cluster centers and the membership grade are updated, and the objective function is minimized to find the best location of the clusters.

$$J\_m = \sum\_{j=1}^{\mathbb{C}} \sum\_{i=1}^{N} \mu\_{ij}^m \left\| \mathbf{x}\_i - \mathbf{c}\_j \right\|^2 \tag{1}$$

where

290 Fuzzy Logic – Emerging Technologies and Applications

d. Other general problems to be considered regarding spatial data are the lack of data or of definition of the object studied, inexact data, inconsistent data from multiple sources, data processing errors, inadequate generalization operations, limitations in the spatial representation scheme and limitations in data acquisition technology (Burroughs, 1996).

In general, cluster analysis involves a set of data in groups or clusters that is organized so that items in the same group are similar to each other and different from those in other groups. In spatial information in clustering, different types of clustering analyses have been studied, including spatial clustering (clustering of spatial points), regionalization (clustering with geographic contiguity constraints) and point pattern analysis (hot-spot detection with spatial scan statistics). The use of many of these techniques for hot-spot detection is relatively problematic for several reasons, including the relatively arbitrary definition of the number of clusters to be included and the procedures applied to draw hot-spot boundaries. These contour areas indicate high to low robbery occurrence and, therefore, respond to the demand for public safety or provide alternatives to precisely locate schools in geographic

Fuzzy clustering methods allow objects to belong to several clusters simultaneously with different degrees of membership; in this work, we used Fuzzy C-Means clustering (FCM), (Bezdek, 1973). FCM is a data clustering technique that considers each data point belonging to a cluster to a certain degree, as specified by a membership degree. Two geospatial models with different applications are presented—hot-spot crime detection and educational

By definition, a hot-spot is a geographic area that presents a greater concentration of events as compared to its surroundings. It is an important tool for the analysis of point data to describe criminal activities, their geospatial distribution, and especially trends—in order to determine zones more likely to have higher concentrations of criminal events. The algorithms utilized to define a hot-spot may vary significantly when determining optimal and representative clusters—i.e., an adequate grouping must be determined. Generally, analysts must examine a series of possible solutions to spatially determine the optimal

The strict assignment of parameters in the hard-clustering algorithm prevents identifying the optimal number of groups and, therefore, the result is not always realistic. Grubesic

In order to disclose spatial crime patterns, Lopez-Caloca et al. (2009) tested criminal spatial patterns in the Mexican city of Hermosillo, as well as moving robberies (vehicle theft and public transportation robbery), fixed robberies (household or commercial establishment robberies) and violent robberies. Geo-referenced data from police records are available for each of the events reported during 2005 and 2006. The advantage of fuzzy clusters is a closer estimation of the delimitation of boundaries based on the information, with which to

configuration; for example, cases using known methods such as hard clustering.

(2006) focuses on fuzzy grouping in the case of delinquency.

**3. Fuzzy spatial clustering** 

distribution plans.

**3.1 Spatial analysis of crime** 

analyze processes in the region.

planning.

dij =ǁ xi-cjǁ2, is the distances of the pattern xi to the cluster centroid cj,

ǁ ǁ is any norm expressing the similarity between any measured data and the center*,* 

*uij* is the degree of membership of *xi* in the cluster *j*,

*xi* is the *i-*th of d-dimensional measured data, *cj* is the centroid of the cluster,

N is the number of data points,

C is the number of clusters, the parameter *m* is the weighted exponent for *uij* and controls the "fuzziness" of the resulting cluster,

*m* is any real number greater than 1 and is called the fuzzifier parameter, for which 2 is usually chosen.

Fuzzy partitioning is carried out through an iterative optimization of the objective function shown above, with the updating of membership *uij* and the cluster centers *cj* by:

$$
\mu\_{ij} = \frac{1}{\sum\_{u=1}^{C} \left( \left\| \mathbf{x}\_i - c\_j \right\| / \left\| \mathbf{x}\_i - c\_u \right\| \right)^{2fm-1}} \tag{2}
$$

$$
\text{where}
\quad c\_j = \frac{\sum\_{i=1}^{N} \mathbf{u}\_{ij}^m \cdot \mathbf{x}\_i}{\sum\_{i=1}^{N} \mathbf{u}\_{ij}^m}
$$

$$
\mathbf{x}\_i \quad \text{and}
$$

The algorithm is comprised of the following steps:

a. Initialize U (membership matrix), called the fuzzy partition matrix, where *uij* denotes the membership degree of a datum xi to cluster *i*,

$$\boldsymbol{U} = \begin{bmatrix} \boldsymbol{\mu}\_{ij} \end{bmatrix} \\ matri \boldsymbol{x}\_{\prime} \boldsymbol{U}^{(0)} ;$$

b. Compute the vectors of the center prototypes

$$\mathbf{C}^{(k)} = \begin{bmatrix} c\_j \end{bmatrix} \text{ with } \boldsymbol{\varPi}^{(k)};$$

c. Compute the distances: <sup>2</sup> *DijA*

Fuzzy Modeling of Geospatial Patterns 293

where uij is the membership of data point j in the cluster, c is the

Classification Entropy (CE). This index tends to increase until it remains at similar values.

where vi is the cluster center of the j-th cluster

Separation Index (S). The more the clusters are separated, the smaller is S, indicating an

Xie and Beni's Index (XB). The optimal number of clusters should minimize the value of

Dunn's Index. To define the optimal number for the cluster, the maximum value for the

*c xy*

Figure 2 (left image) shows a vehicle crime data set for the city of Hermosillo, with the result shown on the city-block network. The right image shows the optimal partition solution, as well as the membership gradient for each cluster. A close-up is represented in Figure 2, where we can see the zones with more crime—streets and areas that need more public safety

 , ,

*k c xy*

Table 1. Different validation measurements proposed in the literature.

max max , *<sup>i</sup> x C*

Partition Coefficient (PC). The optimal number for the cluster corresponds to the maximum PC value. This index tends to decrease, losing a direct connection with the

data.

<sup>1</sup> ( )

( )

*SC c*

( )

the index.

( )

measures.

*XB c*

*S c*

*i*

cluster partitions.

 2

*ij*

2

Partition Index (SC). A lower *SC* value indicates a better partition.

*ij ij*

1 1

1 1 <sup>1</sup> ( ) log *c N*

*i j CE c u u N* 

1

1

*N*

1 ,

min

1

DI must be obtained.

1 ,

min

where d is a distance function

*i ij k i*

*N xv*

*<sup>N</sup> <sup>m</sup> ij j i <sup>C</sup> <sup>j</sup> C*

2 1 1

*N xv*

optimal value for the partition.

*ij j i <sup>C</sup> <sup>j</sup>*

*i ik k i*

( )

*u xv*

min , ( ) min min

*dxy DI c*

*<sup>N</sup> <sup>m</sup> ij j i <sup>C</sup> <sup>j</sup>*

*N xv*

( )

*i ki k*

*ux v*

<sup>2</sup> <sup>2</sup>

*u xv*

2

2

2

*j c j ci j d*

*i j PC c u N*

*C N*

 <sup>2</sup> *<sup>T</sup> D x c Ax c ijA i <sup>j</sup> <sup>i</sup> <sup>j</sup>* , a squared inner-product norm.

Depending on the data and the application, different types of similarity measures may be used to identify classes, where the similarity measure controls how the clusters are formed. Some examples of values that can be used as similarity measures include distance, connectivity, and intensity;


if *k k* <sup>1</sup> *U U* then STOP, otherwise return to step b.

This will stop when <sup>1</sup> max *k k ij ij ij u u* , where is a criterion between 0 and 1 (fuzzy membership), whereas *k* are the iteration steps.

Cluster validity refers to the algorithm problem that attempts to find the best fit for the fixed number of clusters and the parameterized cluster shapes. To perform validity measures, different indexes are calculated that indicate the level of partition (Xie et al. 1991). The indexes calculated are: Partition Coefficient (measures the amount of "overlapping" between clusters), Partition Index (the ratio of the sum of compactness and separation of the clusters), Classification entropy (basically a measurement of the fuzziness of the cluster partition only), Separation Index (the inverse of the partition index measurement), Xie and Beni's Index (aims to quantify the ratio of the total variation within clusters and the separation of clusters) and Dunn's Index (identifies compact and well-separated clusters)( Abonyi et al. 2003 & Balasto B et al. 2003). Table 1 shows the formulas for calculating each index.

The cluster algorithm also attempts to find the best fit for a fixed cluster number and initial conditions; nevertheless, this does not mean that the best fit is significant, since the cluster number could be incorrect. In the case of our data, Figure 1 shows the validation indexes for different fits for the cluster number. The strategy to follow to determine the appropriate cluster number is to calculate a large cluster number and reduce the number based on the data obtained from the validation indexes. It is worth mentioning that each index alone would not be very representative, therfore a set of validations indexes is considered. We take into account that the partitions with fewer groups are better, when the differences between validation values are less. The cluster partition properties are evaluated using PC, CE, SC and the Xie-Beni Index. Cluster properties such as compactness (or variation) and separation are evaluated using the Dunn Index.

Using the data for vehicle crime from 2005-2006 for 437 cases, the PC has a decreasing monotonic trend for C=4, 5 and CE has a monotonically increasing trend. For C=22, the S, SC and XB indexes arrive at their minimum values of de 0.0001, 0.0203 and 0.0025, respectively. The Dunn Index was 2.1038, the determination of the optimal number cluster was primarily based on the SC, DI, S and XB indexes, which affirms that the interpretation using different methods makes it possible to assign an optimal number to clusters.

Partition Coefficient (PC). The optimal number for the cluster corresponds to the maximum PC value. This index tends to decrease, losing a direct connection with the data.

$$PC(\mathbf{c}) = \frac{1}{N} \sum\_{i=1}^{C} \sum\_{j=1}^{N} \left(\mathbf{u}\_{\bar{\eta}}\right)^{2} \text{ where } \mathbf{u}\_{\bar{\eta}} \text{ is the membership of data point j in the cluster, c is the } \bar{\eta}$$

cluster partitions.

292 Fuzzy Logic – Emerging Technologies and Applications

*D x c Ax c ijA i <sup>j</sup> <sup>i</sup> <sup>j</sup>* , a squared inner-product norm.

Depending on the data and the application, different types of similarity measures may be used to identify classes, where the similarity measure controls how the clusters are formed. Some examples of values that can be used as similarity measures include distance,

then STOP, otherwise return to step b.

, where

Cluster validity refers to the algorithm problem that attempts to find the best fit for the fixed number of clusters and the parameterized cluster shapes. To perform validity measures, different indexes are calculated that indicate the level of partition (Xie et al. 1991). The indexes calculated are: Partition Coefficient (measures the amount of "overlapping" between clusters), Partition Index (the ratio of the sum of compactness and separation of the clusters), Classification entropy (basically a measurement of the fuzziness of the cluster partition only), Separation Index (the inverse of the partition index measurement), Xie and Beni's Index (aims to quantify the ratio of the total variation within clusters and the separation of clusters) and Dunn's Index (identifies compact and well-separated clusters)( Abonyi et al.

is a criterion between 0 and 1

<sup>2</sup> *<sup>T</sup>*

(fuzzy membership), whereas *k* are the iteration steps.

*k k ij ij ij u u*

2003 & Balasto B et al. 2003). Table 1 shows the formulas for calculating each index.

The cluster algorithm also attempts to find the best fit for a fixed cluster number and initial conditions; nevertheless, this does not mean that the best fit is significant, since the cluster number could be incorrect. In the case of our data, Figure 1 shows the validation indexes for different fits for the cluster number. The strategy to follow to determine the appropriate cluster number is to calculate a large cluster number and reduce the number based on the data obtained from the validation indexes. It is worth mentioning that each index alone would not be very representative, therfore a set of validations indexes is considered. We take into account that the partitions with fewer groups are better, when the differences between validation values are less. The cluster partition properties are evaluated using PC, CE, SC and the Xie-Beni Index. Cluster properties such as compactness (or variation) and

Using the data for vehicle crime from 2005-2006 for 437 cases, the PC has a decreasing monotonic trend for C=4, 5 and CE has a monotonically increasing trend. For C=22, the S, SC and XB indexes arrive at their minimum values of de 0.0001, 0.0203 and 0.0025, respectively. The Dunn Index was 2.1038, the determination of the optimal number cluster was primarily based on the SC, DI, S and XB indexes, which affirms that the interpretation using different methods makes it possible to assign an optimal number to

connectivity, and intensity;

e. Iteration stop

d. Update the partition matrix *k U* , *<sup>k</sup>* <sup>1</sup> *U* ;

if *k k* <sup>1</sup> *U U*

This will stop when <sup>1</sup> max

separation are evaluated using the Dunn Index.

clusters.

Classification Entropy (CE). This index tends to increase until it remains at similar values.

$$CE(c) = -\frac{1}{N} \sum\_{i=1}^{c} \sum\_{j=1}^{N} \mu\_{ij} \log \left( \mu\_{ij} \right),$$

Partition Index (SC). A lower *SC* value indicates a better partition.

$$\text{SCC}(\mathbf{c}) = \sum\_{i=1}^{\mathbb{C}} \frac{\mathbf{u}\_{ij}^{m} \left\| \mathbf{x}\_{j} - \boldsymbol{\upsilon}\_{i} \right\|^{2}}{N\_{i} \sum\_{k=1}^{\mathbb{C}} \left\| \mathbf{x}\_{k} - \boldsymbol{\upsilon}\_{i} \right\|^{2}} \text{ where } \mathbf{v}\_{i} \text{ is the cluster center of the j-th cluster.}$$

Separation Index (S). The more the clusters are separated, the smaller is S, indicating an optimal value for the partition.

$$S(c) = \sum\_{i=1}^{\mathbb{C}} \frac{\sum\_{j=1}^{N} (\mu\_{ij})^2 \left\| \mathbf{x}\_j - \boldsymbol{\upsilon}\_i \right\|^2}{N \min\_{i,k} \left\| \mathbf{x}\_k - \boldsymbol{\upsilon}\_i \right\|^2}$$

Xie and Beni's Index (XB). The optimal number of clusters should minimize the value of the index.

$$\text{XB}(\mathbf{c}) = \sum\_{i=1}^{\mathbb{C}} \frac{\sum\_{j=1}^{N} (\mu\_{ij})^m \left\| \mathbf{x}\_j - \boldsymbol{\upsilon}\_i \right\|^2}{N \min\_{i,j} \left\| \mathbf{x}\_k - \boldsymbol{\upsilon}\_i \right\|^2}$$

Dunn's Index. To define the optimal number for the cluster, the maximum value for the DI must be obtained.

$$DI(c) = \min\_{j \ge c} \left\{ \min\_{j \ge c, i \ne j} \left\{ \frac{\min\_{x \in C\_i} d(x, y)}{\max\_{k \ge c} \left\{ \max\_{x, y \in} c^d(x, y) \right\}} \right\} \right\}$$

where d is a distance function

Table 1. Different validation measurements proposed in the literature.

Figure 2 (left image) shows a vehicle crime data set for the city of Hermosillo, with the result shown on the city-block network. The right image shows the optimal partition solution, as well as the membership gradient for each cluster. A close-up is represented in Figure 2, where we can see the zones with more crime—streets and areas that need more public safety measures.

Fuzzy Modeling of Geospatial Patterns 295

The crime event is dynamic, thus a more detailed study is needed to consider the temporal portion of the data (months, weeks and days). The membership geovisualization method used was a Sammon mapping (Sammon, 1969), which preserves inter-pattern distances

*sammon*

*S*

project is nonlinear and the stress function is defined as:

*i j*

*d D*

1 *ij ij*

*ij i j ij*

2

*d d*

where dij represents the proximity of point data i and j in the original data space and Dij represents the Euclidian distance between mapped points i and j in the projected space. The

*d D*

Figure 3 shows the geovisualization of the data, where the map's contours are drawn using the selection of membership groups with similar partition values. The areas with greater

*i j ij*

*d*

Fig. 3. Geo-projection result of Sammon's mapping. Close-up of hot-spots generated for vehicle crime during 2005-2006. The hot-spot map identifies places and neighborhoods needing measures to resolve public safety concerns, with different degrees of urgency.

*ij ij*

2

using the Euclidian interpoint distance norm.

Fig. 1. These graphs show the PC, CE, SC, DI, S and XB indexes. The analysis of all of these makes it possible to determine how many clusters are to be represented.

Fig. 2. These figures show the city of Hermosillo Mexico. The left image shows a data set related to vehicle crime; the right image shows the result obtained, of 22 clusters. Data are displayed over the city-block network.

Fig. 1. These graphs show the PC, CE, SC, DI, S and XB indexes. The analysis of all of these

Fig. 2. These figures show the city of Hermosillo Mexico. The left image shows a data set related to vehicle crime; the right image shows the result obtained, of 22 clusters. Data are

displayed over the city-block network.

makes it possible to determine how many clusters are to be represented.

The crime event is dynamic, thus a more detailed study is needed to consider the temporal portion of the data (months, weeks and days). The membership geovisualization method used was a Sammon mapping (Sammon, 1969), which preserves inter-pattern distances using the Euclidian interpoint distance norm.

$$S\_{\text{sammon}} = \frac{1}{\sum\_{i$$

where dij represents the proximity of point data i and j in the original data space and Dij represents the Euclidian distance between mapped points i and j in the projected space. The project is nonlinear and the stress function is defined as:

$$\sum\_{i$$

Figure 3 shows the geovisualization of the data, where the map's contours are drawn using the selection of membership groups with similar partition values. The areas with greater

Fig. 3. Geo-projection result of Sammon's mapping. Close-up of hot-spots generated for vehicle crime during 2005-2006. The hot-spot map identifies places and neighborhoods needing measures to resolve public safety concerns, with different degrees of urgency.

Fuzzy Modeling of Geospatial Patterns 297

The membership zones obtained with fuzzy clusters indicate that 13 areas are benefited by schools. The validation of the optimal clusters is shown in Figure 5. PC has a decreasing monotonic trend and CE has an increasing monotonic trend, representing the increase in the cluster number. In both cases, the connection with the data structure is not direct. SC, S and XB have values of 0.0003, 0.0341 and 1.8785, respectively, and arrive at a local minimum. The DI has a maximum value of 0.524 and affirms the number C=13. With the different

Fig. 5. Graphic representation of the validation index values used to find the optimal cluster

To identify service offerings and demand, and whether they are sufficient, the spatial distribution of the population density was analyzed. The 3 highest membership levels next to 1 were determined for the number of children between 6 and 12 years of age located in this zone. The results show the fact that the educational establishments are not spatially distributed according to the population density of these children, and that the potential population of children between 1 and 4 years of age calls for increasing these services in the future. Zones with low membership are sectors where measures could be taken to avoid this school deficit (Table 2). The results shown in Table 2 indicate that this type of analysis enables defining the geospatial problematic at the neighborhood as well as the street level. The FL tool is fundamental to strategies for locating elementary schools to improve services. A study of this type must take into account the number allotments offered by the

The area of influence of an educational establishment does not necessarily have a circumferential shape. But the existence of topographic and geographic borders must also be

methods of analysis, 13 is chosen as the optimal cluster number.

number.

educational institution.

membership are assigned a value of 1. As the values move away from the ideal value or the center of the set, decreasing values are assigned on a continuous scale from 1 to 0. The values found in the transition zone are shown as intermediate contours. It can be seen that there are up to four blocks in one cluster (representation of the urban region of the city of Hermosillo). Considering the transition contours and regional representations related to stolen vehicles enables defining better strategies to address the problem. The figure numerically shows the monitoring zones according to different degrees of urgency.

In practice, data for different criminal acts can occur at any time; the data are dynamic and changing. Working with fuzzy hot-spot information makes it possible to consider a spatial distribution with grades of membership, enabling administrators and professionals in crime prevention to use the data as a detection strategy as well as to spatially identify different priority zones, taking into greater account urban geographic spaces.

#### **3.2 School infrastructure analysis**

One factor in the level of development of a society is the degree of education. The topic of education has two spatial aspects: first, the identification of zones covered by elementary schools (primary and secondary school categories) and second, the identification of the deficit, based on the comparison of the number of allotments offered versus the population density throughout the zone.

Figure 4 shows the areas in terms of the presence of private and public elementary schools in the Alvaro Obregon district of Mexico City. In fuzzy clustering, the data points for the primary location may belong to more than one cluster, and associated with each of the points are membership grades indicating the degree to which the data points belong to the different clusters.

Fig. 4. The Alvaro Obregon district in Mexico City. Zones with elementary school coverage.

membership are assigned a value of 1. As the values move away from the ideal value or the center of the set, decreasing values are assigned on a continuous scale from 1 to 0. The values found in the transition zone are shown as intermediate contours. It can be seen that there are up to four blocks in one cluster (representation of the urban region of the city of Hermosillo). Considering the transition contours and regional representations related to stolen vehicles enables defining better strategies to address the problem. The figure

In practice, data for different criminal acts can occur at any time; the data are dynamic and changing. Working with fuzzy hot-spot information makes it possible to consider a spatial distribution with grades of membership, enabling administrators and professionals in crime prevention to use the data as a detection strategy as well as to spatially identify different

One factor in the level of development of a society is the degree of education. The topic of education has two spatial aspects: first, the identification of zones covered by elementary schools (primary and secondary school categories) and second, the identification of the deficit, based on the comparison of the number of allotments offered versus the population

Figure 4 shows the areas in terms of the presence of private and public elementary schools in the Alvaro Obregon district of Mexico City. In fuzzy clustering, the data points for the primary location may belong to more than one cluster, and associated with each of the points are membership grades indicating the degree to which the data points belong to the

Fig. 4. The Alvaro Obregon district in Mexico City. Zones with elementary school coverage.

numerically shows the monitoring zones according to different degrees of urgency.

priority zones, taking into greater account urban geographic spaces.

**3.2 School infrastructure analysis** 

density throughout the zone.

different clusters.

The membership zones obtained with fuzzy clusters indicate that 13 areas are benefited by schools. The validation of the optimal clusters is shown in Figure 5. PC has a decreasing monotonic trend and CE has an increasing monotonic trend, representing the increase in the cluster number. In both cases, the connection with the data structure is not direct. SC, S and XB have values of 0.0003, 0.0341 and 1.8785, respectively, and arrive at a local minimum. The DI has a maximum value of 0.524 and affirms the number C=13. With the different methods of analysis, 13 is chosen as the optimal cluster number.

Fig. 5. Graphic representation of the validation index values used to find the optimal cluster number.

To identify service offerings and demand, and whether they are sufficient, the spatial distribution of the population density was analyzed. The 3 highest membership levels next to 1 were determined for the number of children between 6 and 12 years of age located in this zone. The results show the fact that the educational establishments are not spatially distributed according to the population density of these children, and that the potential population of children between 1 and 4 years of age calls for increasing these services in the future. Zones with low membership are sectors where measures could be taken to avoid this school deficit (Table 2). The results shown in Table 2 indicate that this type of analysis enables defining the geospatial problematic at the neighborhood as well as the street level.

The FL tool is fundamental to strategies for locating elementary schools to improve services. A study of this type must take into account the number allotments offered by the educational institution.

The area of influence of an educational establishment does not necessarily have a circumferential shape. But the existence of topographic and geographic borders must also be

Fuzzy Modeling of Geospatial Patterns 299

great that establishing one was considered beneficial to the local community. Identifying the precise location of the school is now important. Educational planning requires a geographic

Though the concept of data fusion is easy to understand, it varies from study to another. Data fusion has also been referred to as merging, combination, synergy and integration. In terms of formal data fusion tasks, it is desirable to design an architecture that combines information from different sources, thus obtaining high-quality information. Current fusion methods utilize tools such as weighted average, neural networks (multi-sensory fusion), rules-based knowledge, wavelets (multiresolution fusion), graph pyramids, and more recently, fuzzy logic. Data fusion is frequently described in literature as occurring on three levels: pixel, attribute and decision (Pohl et al. 1998, Wald 2002, López-Caloca 2006). A general idea about how different authors handle these fusion levels is described below.

a. Pixel level. Images come from different sources, which are combined from pixel to pixel. The fusion process should preserve the relevant information from the entered images on the synthetic image (pattern preservation). Although the word "pixel" is not really adequate, the pixel is the basis for the information and does not have semantic

b. Attribute level. The figures (geometric, structural or spectral) are drawn from crude images and fused afterwards. Fusion at the figure level requires recognizable objects extracted from diverse data sources using a segmentation process. The figures correspond to characteristics extracted from the initial images, such that they provide form; selection is based on the practical use of the application. The classified maps are combined and the spacial information related to each pixel's neighbor is taken into

c. Decision level. Decision fusion can be defined as the process of fusing information from several individual data sources after each data source has undergone a preliminary classification. The results of classification are combined by using their weighted

For hard classifiers, image pixels are assigned to a given category, although errors in pixel classifications exist (pixels that may belong to a different category). When assigning a pixel to a class, there is a risk of it being assigned to a class to which it does not belong (misclassified) or pixels may be over-classified. Fuzzy classification considers that one category admits a property between 0 and 1. The idea is to permit simultaneous assignment to various categories

The fuzzy classification problem has been extensively studied in remote sensing (Lizarazo & Elsner, 2011, Amici et al. ,2004). A fuzzy classifier is mainly applied when the data have a high degree of spectral mixture. Shackelford and Davis (2003) present a fuzzy logic classifier and object-based approach. The individual pixels in the image are first classified with a fuzzy classifier, making use of both spectral and spatial information. The segmented image is then used with additional object feature information to classify the image objects. Huntsherger (1985) described the application of the technique, called iterative fuzzy clustering, with the aim that the segmentation process not be affected by noise and

with different degrees of property, and later reclassify the fuzzy boundaries.

distribution study of the current density of schools.

**4. Fuzzy-based data fusion model** 

account in order to improve the fit.

meaning.

significance.

considered because they sometimes constitute significant limitations in identifying a student's route to his or her school. A diagnostic for the student's travel distance network is shown by Reyes-Guerrero (1986), who defines the degree of membership in a class for each one of the nodes on the graph (nodes represent the elements that belongs to each region), making it possible to define neighborhood relationships.

Figure 6 represents the analysis of distance from cluster centers, considering which of the clusters with greater school density are in closer proximity to each other. This analysis makes it possible to prevent excessive proximity in order to avoid, as much as possible, a spatially inadequate school distribution. In fact, the demand for elementary schools was so


Table 2. Distribution of educational services for children between 6 and 12 years of age.

Fig. 6. Example of the distance analysis based on cluster centers; zones with less membership represent those with low elementary school coverage for the population.

considered because they sometimes constitute significant limitations in identifying a student's route to his or her school. A diagnostic for the student's travel distance network is shown by Reyes-Guerrero (1986), who defines the degree of membership in a class for each one of the nodes on the graph (nodes represent the elements that belongs to each region),

Figure 6 represents the analysis of distance from cluster centers, considering which of the clusters with greater school density are in closer proximity to each other. This analysis makes it possible to prevent excessive proximity in order to avoid, as much as possible, a spatially inadequate school distribution. In fact, the demand for elementary schools was so

Table 2. Distribution of educational services for children between 6 and 12 years of age.

Fig. 6. Example of the distance analysis based on cluster centers; zones with less membership represent those with low elementary school coverage for the population.

Distribution of educational services. (Alvaro Obregon, Mexico City)

Molino de Rosas-Mixcoac; Molino de Santo Domingo-Acueducto; Olivar de los Padres; Piloto Adolfo López Mateos; Pueblo Santa Fe-Gamitos; San Angel –Pogreso; San Bartolo Ameyalco; Torres de Potrero; Alfonso XIII; Bosques 1A-2A seccion; Ceguayo-Cuevitas; Jardines del Pedregal- CU; Jardines del Pedregal-Loreto

Presidentes - Golondrinas – Lomas de Capula; Colinas de Tarango – Lomas de Tarango; San Clemente norte y sur; Cedros

making it possible to define neighborhood relationships.

Areas with current educational services

Areas with deficit of educational services

great that establishing one was considered beneficial to the local community. Identifying the precise location of the school is now important. Educational planning requires a geographic distribution study of the current density of schools.

#### **4. Fuzzy-based data fusion model**

Though the concept of data fusion is easy to understand, it varies from study to another. Data fusion has also been referred to as merging, combination, synergy and integration. In terms of formal data fusion tasks, it is desirable to design an architecture that combines information from different sources, thus obtaining high-quality information. Current fusion methods utilize tools such as weighted average, neural networks (multi-sensory fusion), rules-based knowledge, wavelets (multiresolution fusion), graph pyramids, and more recently, fuzzy logic. Data fusion is frequently described in literature as occurring on three levels: pixel, attribute and decision (Pohl et al. 1998, Wald 2002, López-Caloca 2006). A general idea about how different authors handle these fusion levels is described below.


For hard classifiers, image pixels are assigned to a given category, although errors in pixel classifications exist (pixels that may belong to a different category). When assigning a pixel to a class, there is a risk of it being assigned to a class to which it does not belong (misclassified) or pixels may be over-classified. Fuzzy classification considers that one category admits a property between 0 and 1. The idea is to permit simultaneous assignment to various categories with different degrees of property, and later reclassify the fuzzy boundaries.

The fuzzy classification problem has been extensively studied in remote sensing (Lizarazo & Elsner, 2011, Amici et al. ,2004). A fuzzy classifier is mainly applied when the data have a high degree of spectral mixture. Shackelford and Davis (2003) present a fuzzy logic classifier and object-based approach. The individual pixels in the image are first classified with a fuzzy classifier, making use of both spectral and spatial information. The segmented image is then used with additional object feature information to classify the image objects. Huntsherger (1985) described the application of the technique, called iterative fuzzy clustering, with the aim that the segmentation process not be affected by noise and

Fuzzy Modeling of Geospatial Patterns 301

posterior probabilities from the outputs of the neural network and the membership degrees for the fuzzy classifier; i.e., the methodology consists of processing the data with each classifier alone and assigning to the algorithms each pixel's grade of membership for the classes considered. Then, the combination rule from the fuzzy decision is utilized to combine the results furnished by the algorithms, in accordance with the capabilities of the classifiers used. When modeling the output classifier, such as a fuzzy set, certainty is measured by the grade of uncertainty and the estimates of the global exactness of each classifier. The results can integrate a good deal of complementary information for the final

Figure 7 shows a proposed fuzzy fusion technique. Images were classified with the Support Vector Machine (SVM) algorithm. In SVM, a function set is analyzed with this classifier, in such a manner that the function is approximated with less discrepancy between the a priori knowledge and the training data. Classes are divided in feature space; SVM separates two class sets by means of a hyper-plane (H) of linear or nonlinear functions. Available kernels include linear, polynomial, radial basis function and sigmoid. The kernel transformation allows for finding a new feature space in which linear hyper-planes are appropriate for class separation. In order to avoid or minimize the former errors, at the moment of adjusting the data it searches for the Structural Risk Minimization (SRM). In this work, two kernels were utilized—sigmoid and polynomial. The literature reports that both segmentation results present high precision. For the purposes of this task, the fusion of both segmentation results and the application of our proposed fuzzy fusion techniques are proposed. The resulting SVM classifications furnish redundant and complementary results. The methodology consists of data processing with each classifier and assigning each pixel's membership grade to the algorithms for the classes being considered; these are the inputs in the fusion process. The classified images were re-mapped into membership values ranging from 0 to 1, using a

The fuzzy fusion technique is used to combine two or more fuzzy membership results using fuzzy as a simple operator to create, in the case of suitability, the most suitable model. The

classification process.

Fig. 7. Experimental fuzzy fusion scheme

specified fuzzy function; in this case, a linear transformation.

degradation from image acquisition. Likewise, combinations such as fuzzy-support vector machine (F2-SVM) (Borasca, 2006) enable demodulating the relations between one pattern and the proposed classes in the F2-SVM framework. Other classification approaches attempt to take advantage of the strengths of each algorithm. For example, in the combination of two techniques—fuzzy topology and the Maximum Likelihood Classifier (MLC) (Liu et al., 2011), known as FTMLC—one membership function is created for each pixel using FTMLC and the pixels with greater membership are assigned a certain class, while those with less membership are left at the boundaries for a later process. Connectivity is sought for pixels at the boundaries with respect to their 8 neighbors, in such a manner that the one with the higher number of connected pixels belongs to that class. As a result, pixels on fuzzy borders are re-classified and, therefore, are given a higher assignment.

In the search for better solutions to problems of imprecise information, data fusion emerges as an alternate tool which, for example, can use the strengths of different classifiers in order to obtain a better approximation, with the resulting classification proportions resulting in less redundancy and complementary information.

The aggregation of information from multiple sources using a fuzzy system requires specifying the value of the input variable, membership functions and production rules (Klein, 2004, Raol, 2011). Each data source furnishes one or various admissions. An expert develops the standards specifying the outlet actions in terms of fuzzy sets, combinations of fuzzy input, and the definition of property functions that define the property of the fuzzy sets for output.

Fusion tasks at the pixel level have applied FL and neural fuzzy algorithms (Zhao et al., 2005, Meitzler, 2002, Singh, 2004). Their implementation considers two or more admission images for the fuzzy method. The implementation is carried out by assigning the admission variables with the same image size, deciding the number and type of functions for membership to the admission images, applying the fuzzy action using the rules developed in the pixel values of each admission image—which provides a fuzzy set represented by a membership function—and, lastly, applying the defuzzification of the outlet image.

In the fusion framework at the decision level, fuzzy algorithms have also been successfully used in various applications. Chanussot et al. (1999) propose a variety of strategies to combine images based on fuzzy fusion techniques with the aim of drawing roads. As in the case of fuzzy modeling, they combine the results from various detectors of boundaries. The neural-fuzzy-fusion method (NFF) combines a set of fuzzy classifiers in a system called a multiple classifier system (MCS). The application of this method to remote sensing images has demonstrated that the NFF-MCS produces good results (Shankar et al., 2006).

Support vector machines (SVM) have been applied to different classification problems (Mounrakis et al., 2010). The precision of these generally surpasses conventional algorithms. Fauvel et al. (2006, 2007) conducted fusion processes by combining spectral and spatial information. While the SVM enables working with the spectral information of an image, the spatial information is defined by means of morphological profiles, with the fusion process performed using different voting schemes (for example, absolute maximum and majority voting). Mathieu Fauvel et al. (2007) discuss the optimization of classifications of urban zones with high-resolution images, considering the use of various classifiers (conjugate gradient neural network and a fuzzy classifier). The inputs for the fusion process were the

degradation from image acquisition. Likewise, combinations such as fuzzy-support vector machine (F2-SVM) (Borasca, 2006) enable demodulating the relations between one pattern and the proposed classes in the F2-SVM framework. Other classification approaches attempt to take advantage of the strengths of each algorithm. For example, in the combination of two techniques—fuzzy topology and the Maximum Likelihood Classifier (MLC) (Liu et al., 2011), known as FTMLC—one membership function is created for each pixel using FTMLC and the pixels with greater membership are assigned a certain class, while those with less membership are left at the boundaries for a later process. Connectivity is sought for pixels at the boundaries with respect to their 8 neighbors, in such a manner that the one with the higher number of connected pixels belongs to that class. As a result, pixels on fuzzy borders

In the search for better solutions to problems of imprecise information, data fusion emerges as an alternate tool which, for example, can use the strengths of different classifiers in order to obtain a better approximation, with the resulting classification proportions resulting in

The aggregation of information from multiple sources using a fuzzy system requires specifying the value of the input variable, membership functions and production rules (Klein, 2004, Raol, 2011). Each data source furnishes one or various admissions. An expert develops the standards specifying the outlet actions in terms of fuzzy sets, combinations of fuzzy input, and the definition of property functions that define the property of the fuzzy

Fusion tasks at the pixel level have applied FL and neural fuzzy algorithms (Zhao et al., 2005, Meitzler, 2002, Singh, 2004). Their implementation considers two or more admission images for the fuzzy method. The implementation is carried out by assigning the admission variables with the same image size, deciding the number and type of functions for membership to the admission images, applying the fuzzy action using the rules developed in the pixel values of each admission image—which provides a fuzzy set represented by a

In the fusion framework at the decision level, fuzzy algorithms have also been successfully used in various applications. Chanussot et al. (1999) propose a variety of strategies to combine images based on fuzzy fusion techniques with the aim of drawing roads. As in the case of fuzzy modeling, they combine the results from various detectors of boundaries. The neural-fuzzy-fusion method (NFF) combines a set of fuzzy classifiers in a system called a multiple classifier system (MCS). The application of this method to remote sensing images

Support vector machines (SVM) have been applied to different classification problems (Mounrakis et al., 2010). The precision of these generally surpasses conventional algorithms. Fauvel et al. (2006, 2007) conducted fusion processes by combining spectral and spatial information. While the SVM enables working with the spectral information of an image, the spatial information is defined by means of morphological profiles, with the fusion process performed using different voting schemes (for example, absolute maximum and majority voting). Mathieu Fauvel et al. (2007) discuss the optimization of classifications of urban zones with high-resolution images, considering the use of various classifiers (conjugate gradient neural network and a fuzzy classifier). The inputs for the fusion process were the

membership function—and, lastly, applying the defuzzification of the outlet image.

has demonstrated that the NFF-MCS produces good results (Shankar et al., 2006).

are re-classified and, therefore, are given a higher assignment.

less redundancy and complementary information.

sets for output.

posterior probabilities from the outputs of the neural network and the membership degrees for the fuzzy classifier; i.e., the methodology consists of processing the data with each classifier alone and assigning to the algorithms each pixel's grade of membership for the classes considered. Then, the combination rule from the fuzzy decision is utilized to combine the results furnished by the algorithms, in accordance with the capabilities of the classifiers used. When modeling the output classifier, such as a fuzzy set, certainty is measured by the grade of uncertainty and the estimates of the global exactness of each classifier. The results can integrate a good deal of complementary information for the final classification process.

Fig. 7. Experimental fuzzy fusion scheme

Figure 7 shows a proposed fuzzy fusion technique. Images were classified with the Support Vector Machine (SVM) algorithm. In SVM, a function set is analyzed with this classifier, in such a manner that the function is approximated with less discrepancy between the a priori knowledge and the training data. Classes are divided in feature space; SVM separates two class sets by means of a hyper-plane (H) of linear or nonlinear functions. Available kernels include linear, polynomial, radial basis function and sigmoid. The kernel transformation allows for finding a new feature space in which linear hyper-planes are appropriate for class separation. In order to avoid or minimize the former errors, at the moment of adjusting the data it searches for the Structural Risk Minimization (SRM). In this work, two kernels were utilized—sigmoid and polynomial. The literature reports that both segmentation results present high precision. For the purposes of this task, the fusion of both segmentation results and the application of our proposed fuzzy fusion techniques are proposed. The resulting SVM classifications furnish redundant and complementary results. The methodology consists of data processing with each classifier and assigning each pixel's membership grade to the algorithms for the classes being considered; these are the inputs in the fusion process. The classified images were re-mapped into membership values ranging from 0 to 1, using a specified fuzzy function; in this case, a linear transformation.

The fuzzy fusion technique is used to combine two or more fuzzy membership results using fuzzy as a simple operator to create, in the case of suitability, the most suitable model. The

Fuzzy Modeling of Geospatial Patterns 303

This approach enables combining objects extracted after the segmentation process, improving the class boundaries and thereby having better knowledge of the objects observed when all the information is not originally available. The method was also applied

Fig. 9. A) and B): Images with less information are shown inside circular areas with less

We mentioned previously the interest in the fusion of images using fuzzy tools and applying them to remote sensing images. Fuzzy fusion techniques enable alleviating the problem of improving real and complex classifications as well as improving and complementing information. This approach shows the fusion of two SVM classifications by applying the sigmoid and polynomial kernels to a SPOT image with a landscape representative of an agricultural area. These results are shown in Figure 9B and 9C, with the

information, C) reference image and D) image fusion result.

fusion shown in 9D.

with SPOT-5 satellite images.

fuzzy sets A and B will return the minimum value of the sets of cells located at the standard intersection (*A* ∩ *B*)(*x*) = min [*A*(*x*), *B*(*x*)] . Finally, the defuzzification action is conducted by assigning the segmented regions to each class. Defuzzification is a process that converts a fuzzy set or fuzzy number into a crisp value or number.

The proposed method represented in Figure 7 is applied using images in Figure 8. For the purpose of evaluating the combination of elements extracted during the segmentation process, segmentation with SVM (sigmoid) was applied to images 8B and 8C. These images consist of information with little definition, but include complementary information. Fuzzy fusion was applied later, and the result was evaluated with the segmentation of reference image 8A.

Fig. 8. Original images. A) reference, B) Image degraded in the leopard's extremities and C) image degraded in the face of the leopard. (Image example, Barnea & Hassner, 2006).

All images were segmented into 6 classes (Figure 9) that define the background of the image and the leopard shape (spots, face, skin), where image C is a classified reference image, A and B are classified results of the images with lack of sharpness and information, image D is a result of applying the fusion images from A and B using the fuzzy methodology. The segmentation accuracy (SA) was calculated, which is defined as the percentage of the number of correctly classified pixels to the total number of pixels. The well-classified pixels were considered using reference image C. The SA for image A was 89.60%; the SA for image B, 83.12%, and finally, the SA for the fusion image D was 87.14%.

fuzzy sets A and B will return the minimum value of the sets of cells located at the standard intersection (*A* ∩ *B*)(*x*) = min [*A*(*x*), *B*(*x*)] . Finally, the defuzzification action is conducted by assigning the segmented regions to each class. Defuzzification is a process that converts a

The proposed method represented in Figure 7 is applied using images in Figure 8. For the purpose of evaluating the combination of elements extracted during the segmentation process, segmentation with SVM (sigmoid) was applied to images 8B and 8C. These images consist of information with little definition, but include complementary information. Fuzzy fusion was applied later, and the result was evaluated with the segmentation of reference

Fig. 8. Original images. A) reference, B) Image degraded in the leopard's extremities and C) image degraded in the face of the leopard. (Image example, Barnea & Hassner, 2006).

All images were segmented into 6 classes (Figure 9) that define the background of the image and the leopard shape (spots, face, skin), where image C is a classified reference image, A and B are classified results of the images with lack of sharpness and information, image D is a result of applying the fusion images from A and B using the fuzzy methodology. The segmentation accuracy (SA) was calculated, which is defined as the percentage of the number of correctly classified pixels to the total number of pixels. The well-classified pixels were considered using reference image C. The SA for image A was 89.60%; the SA for image B, 83.12%, and finally, the SA for the fusion image D was 87.14%.

fuzzy set or fuzzy number into a crisp value or number.

image 8A.

This approach enables combining objects extracted after the segmentation process, improving the class boundaries and thereby having better knowledge of the objects observed when all the information is not originally available. The method was also applied with SPOT-5 satellite images.

Fig. 9. A) and B): Images with less information are shown inside circular areas with less information, C) reference image and D) image fusion result.

We mentioned previously the interest in the fusion of images using fuzzy tools and applying them to remote sensing images. Fuzzy fusion techniques enable alleviating the problem of improving real and complex classifications as well as improving and complementing information. This approach shows the fusion of two SVM classifications by applying the sigmoid and polynomial kernels to a SPOT image with a landscape representative of an agricultural area. These results are shown in Figure 9B and 9C, with the fusion shown in 9D.

Fuzzy Modeling of Geospatial Patterns 305

the other SVM classifiers (sigmoid and polynomial kernel), which obtain uniformity values of (0.840). One last measurement that can be performed with the error of the results is the verification process, by calculating overall reliability. Photo interpretation was used, such as real land—as well as the verification of maps classified based on the definition of proposed thematic categories (agriculture fields, bare land, among others). The overall reliability was 84.8% for SVM (sigmoid), 83.9% for SVM (polynomial), and lastly, 84.4% for the fuzzy technique. The results indicate that fuzzy fusion has an acceptable thematic quality and may be an alternative to integrate information, in this example, and to obtain well-defined

This section illustrates a simple fusion model with the application of fuzzy concepts, shows its function in providing complementary information (Figure 9) and presents another example applied to geospatial data (Figure 10), such as thematic classification maps obtained from a single satellite image, which can be combined to reduce uncertainty. With this example, we demonstrate that fusion along with fuzzy techniques make it possible to

This work demonstrated the advantages of utilizing fuzzy methods for spatial analysis and image processing applications. For crime analysis, we were able to identify patterns by looking at the geography of the incidents and identifying hot-spots. Zones with a high spatial concentration of schools were also identified, as well as the existence of geographic areas needing this service. Finally, the application of fuzzy fusion enabled combining information within the framework of fuzzy modeling in order to improve and complete

It is therefore possible to conclude from these examples that the models designed and applied in this work allowed us to identify different aspects of spatial patterns, where the

For geospatial analysis, the challenge to explore more applications with fuzzy methodologies continues to evolve. In the next phase, other elements of geospatial structures could naturally be explored, such as the causes of certain phenomena in the regions where crime occurs, in educational planning or in the functioning of social urban processes. The generation of robust scientific knowledge is needed in order to address problems that in the past have not been

Although several mathematical models have been designed for geospatial applications, topological concepts and geographic neighborhood models using fuzzy set tools have received little attention in the area of modeling. Future work will integrate the idea of fuzzy topology proposed by Reyes-Guerrero (1986) to include the topological space as a new mathematical structure, applying the design of fusion and classification algorithms to

The authors thank to Dr. Elvia Martínez, José Manuel Madrigal, Camilo Caudillo and José

topology, contiguity and the degree of membership to a border or interior region.

information, as is the case when using different classifiers.

possible to study with non-fuzzy computational algorithms.

Luis López, Rafael García for their contributions to this work.

main elements of the study were part of the geographical landscape.

images.

model spatial properties.

**6. Acknowledgment** 

**5. Conclusion** 

In order to conduct quantitative comparisons of the two algorithms, the concept of uniformity (Levine & Nazif, 1985, Cheng-Chia et al. 1997) was applied. This method is applied when a reference image or real data do not exist. Let I be the segmented image and SI the area of the entire image. Ri denotes the set of pixels in region i. The uniformity of a segmentation result is defined by:

$$\mathbf{U} = 1 - \frac{\sum\_{i=1}^{S} \sigma\_i}{K} \tag{3}$$

where S is the number of classes, σ2 denotes *i* within-class variance of the i-th class, K is a normalization factor that limits the maximum value of the measurement to 1. We find that the proposed fuzzy fusion method obtains a similar uniformity value (0.889) with respect to

Fig. 10. A) Original, B) SVM (sigmoid kernel ), C) SVM (polynomial kernel) and D) fuzzy fusion result

In order to conduct quantitative comparisons of the two algorithms, the concept of uniformity (Levine & Nazif, 1985, Cheng-Chia et al. 1997) was applied. This method is applied when a reference image or real data do not exist. Let I be the segmented image and SI the area of the entire image. Ri denotes the set of pixels in region i. The uniformity of a

<sup>1</sup> U 1

Fig. 10. A) Original, B) SVM (sigmoid kernel ), C) SVM (polynomial kernel) and D) fuzzy

 

where S is the number of classes, σ2 denotes *i* within-class variance of the i-th class, K is a normalization factor that limits the maximum value of the measurement to 1. We find that the proposed fuzzy fusion method obtains a similar uniformity value (0.889) with respect to

*S i i K*

(3)

segmentation result is defined by:

fusion result

the other SVM classifiers (sigmoid and polynomial kernel), which obtain uniformity values of (0.840). One last measurement that can be performed with the error of the results is the verification process, by calculating overall reliability. Photo interpretation was used, such as real land—as well as the verification of maps classified based on the definition of proposed thematic categories (agriculture fields, bare land, among others). The overall reliability was 84.8% for SVM (sigmoid), 83.9% for SVM (polynomial), and lastly, 84.4% for the fuzzy technique. The results indicate that fuzzy fusion has an acceptable thematic quality and may be an alternative to integrate information, in this example, and to obtain well-defined images.

This section illustrates a simple fusion model with the application of fuzzy concepts, shows its function in providing complementary information (Figure 9) and presents another example applied to geospatial data (Figure 10), such as thematic classification maps obtained from a single satellite image, which can be combined to reduce uncertainty. With this example, we demonstrate that fusion along with fuzzy techniques make it possible to model spatial properties.

#### **5. Conclusion**

This work demonstrated the advantages of utilizing fuzzy methods for spatial analysis and image processing applications. For crime analysis, we were able to identify patterns by looking at the geography of the incidents and identifying hot-spots. Zones with a high spatial concentration of schools were also identified, as well as the existence of geographic areas needing this service. Finally, the application of fuzzy fusion enabled combining information within the framework of fuzzy modeling in order to improve and complete information, as is the case when using different classifiers.

It is therefore possible to conclude from these examples that the models designed and applied in this work allowed us to identify different aspects of spatial patterns, where the main elements of the study were part of the geographical landscape.

For geospatial analysis, the challenge to explore more applications with fuzzy methodologies continues to evolve. In the next phase, other elements of geospatial structures could naturally be explored, such as the causes of certain phenomena in the regions where crime occurs, in educational planning or in the functioning of social urban processes. The generation of robust scientific knowledge is needed in order to address problems that in the past have not been possible to study with non-fuzzy computational algorithms.

Although several mathematical models have been designed for geospatial applications, topological concepts and geographic neighborhood models using fuzzy set tools have received little attention in the area of modeling. Future work will integrate the idea of fuzzy topology proposed by Reyes-Guerrero (1986) to include the topological space as a new mathematical structure, applying the design of fusion and classification algorithms to topology, contiguity and the degree of membership to a border or interior region.

#### **6. Acknowledgment**

The authors thank to Dr. Elvia Martínez, José Manuel Madrigal, Camilo Caudillo and José Luis López, Rafael García for their contributions to this work.

Fuzzy Modeling of Geospatial Patterns 307

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**Greenhouse Fuzzy and Neuro-Fuzzy** 

Gorrostieta-Hurtado Efren1, Pedraza-Ortega Jesus Carlos1,

Tovar-Arriaga Saúl1 and Sotomayor-Olmedo Artemio2 *1Facultad de Informática, Universidad Autónoma de Querétaro, 2Facultad de Ingeniería, Universidad Autónoma de Querétaro,* 

Aceves-Fernández Marco Antonio, Ramos-Arreguín Juan Manuel1,

During the last decades, a considerable effort was devoted to develop adequate greenhouse climate and crop models, for driving simulation, control and managing (Guzmán-Cruz, *et. Al*, Rico-Garcia, *et al* ). The study and design of greenhouse environmental models implies having a clear understanding of the greenhouse climate processes. These models must be related with the external influences of the outside weather conditions (such as solar radiation, outside air temperature, wind velocity, etc.), and with the control actions performed (such as ventilation, cooling, heating, among others). The practical goal of this work is to model the greenhouse air temperature and humidity using clustering techniques and made an automatically generator of fuzzy rules relations from real data in order to

The soft computing techniques, such as neural networks, clustering algorithms and fuzzy logic, have been successfully applied to classification and pattern recognition. Besides, fuzzy logic is highly used when the system modeling implies information is scarce, imprecise or when the system is described by complex mathematical model. An example of this kind of structure is a greenhouse and it's inherit variables such as: indoor and outdoor temperature and humidity, wind direction and speed, etc. These variables present a dynamic and nonlinear behavior; being the in-house temperature and internal humidity the key variables for the greenhouse control and modeling. In this chapter, the construction of fuzzy systems by fuzzy c-means and fuzzy subtractive clustering are described. Finally a comparison with adaptive neuro-fuzzy inference system (anfis) and neural networks will be presented.

The non-linear behavior of the greenhouse-climate is a combination of complex physical interactions between energy transfer such as radiation and temperature and mass transfer

like humidity and wind (indoor and outdoor the greenhouse).

**1. Introduction** 

predict the behavior inside the greenhouse.

**2. Greenhouse model** 

**Modeling Techniques** 


### **Greenhouse Fuzzy and Neuro-Fuzzy Modeling Techniques**

 Gorrostieta-Hurtado Efren1, Pedraza-Ortega Jesus Carlos1, Aceves-Fernández Marco Antonio, Ramos-Arreguín Juan Manuel1, Tovar-Arriaga Saúl1 and Sotomayor-Olmedo Artemio2 *1Facultad de Informática, Universidad Autónoma de Querétaro, 2Facultad de Ingeniería, Universidad Autónoma de Querétaro, Querétaro, México* 

#### **1. Introduction**

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features. In: P.A. Burrough and A.U. Frank (Eds). *In Geographic objects with* 

Spatial Resolutions. Presses de l'Ecole, Ecole des Mines de Paris, Paris, France,

During the last decades, a considerable effort was devoted to develop adequate greenhouse climate and crop models, for driving simulation, control and managing (Guzmán-Cruz, *et. Al*, Rico-Garcia, *et al* ). The study and design of greenhouse environmental models implies having a clear understanding of the greenhouse climate processes. These models must be related with the external influences of the outside weather conditions (such as solar radiation, outside air temperature, wind velocity, etc.), and with the control actions performed (such as ventilation, cooling, heating, among others). The practical goal of this work is to model the greenhouse air temperature and humidity using clustering techniques and made an automatically generator of fuzzy rules relations from real data in order to predict the behavior inside the greenhouse.

The soft computing techniques, such as neural networks, clustering algorithms and fuzzy logic, have been successfully applied to classification and pattern recognition. Besides, fuzzy logic is highly used when the system modeling implies information is scarce, imprecise or when the system is described by complex mathematical model. An example of this kind of structure is a greenhouse and it's inherit variables such as: indoor and outdoor temperature and humidity, wind direction and speed, etc. These variables present a dynamic and nonlinear behavior; being the in-house temperature and internal humidity the key variables for the greenhouse control and modeling. In this chapter, the construction of fuzzy systems by fuzzy c-means and fuzzy subtractive clustering are described. Finally a comparison with adaptive neuro-fuzzy inference system (anfis) and neural networks will be presented.

#### **2. Greenhouse model**

The non-linear behavior of the greenhouse-climate is a combination of complex physical interactions between energy transfer such as radiation and temperature and mass transfer like humidity and wind (indoor and outdoor the greenhouse).

Greenhouse Fuzzy and Neuro-Fuzzy Modeling Techniques 311




The working of FIS is as follows. The inputs are converted in to fuzzy by using fuzzification method. After fuzzification the rule base is formed. The rule base and the database are

Defuzzification is used to convert fuzzy value to the real world value which is the output. The steps of fuzzy reasoning (inference operations upon fuzzy IF–THEN rules) performed

 Compare the input variables with the membership functions on the antecedent part to obtain the membership values of each linguistic label. (this step is often called

 Combine (through a specific t-norm operator, usually multiplication or min) the membership values on the premise part to get firing strength (weight) of each rule. Generate the qualified consequents (either fuzzy or crisp) or each rule depending on the

Aggregate the qualified consequents to produce a crisp output. (This step is called

There are a number of fuzzy clustering techniques available. In this work, two fuzzy clustering methods have been chosen: fuzzy c-means clustering and fuzzy clustering subtractive algorithms. These methods are proven to be the most reliable fuzzy clustering methods as well as better forecasters in terms of absolute error according to some

Since 1985 when the fuzzy model methodology suggested by Takagi-Sugeno [Takagi *et al*  1985, Sugeno *et al* 1988], as well known as the TSK model, has been widely applied on

Fuzzy system needs the precedent and consequence to express the logical connection between the input output datasets that are used as a basis to produce the desired system

Fuzzy C-Means clustering (FCM) is an iterative optimization algorithm that minimizes the

theoretical analysis, control applications and fuzzy modeling.


The function of each block is as follows:

jointly referred to as the knowledge base.

linguistic values; and

crisp output.

fuzzification.)

firing strength.

defuzzification.)

authors[Sin, Gomez, Chiu].

behavior [Sin *et al* 1993].

cost function given by:

**4.1 Fuzzy Clustering Means (FCM)** 

**4. Fuzzy clustering techniques** 

rules;

by FISs are:


In this work the humidity and temperature are considered as the greenhouse key parameters, based on (Guzmán-Cruz, *et. Al*, Rico-Garcia, *et al*) observations.

#### **3. Fuzzy systems**

Fuzzy inference systems (FIS) are also known as fuzzy rule-based systems. This is a major unit of a fuzzy logic system. The decision-making is an important part in the entire system. The FIS formulates suitable rules and based upon the rules the decision is made. This is mainly based on the concepts of the fuzzy set theory, fuzzy IF–THEN rules, and fuzzy reasoning. FIS uses "IF - THEN" statements, and the connectors present in the rule statement are "OR" or "AND" to make the necessary decision rules.

Fuzzy inference system consists of a fuzzification interface, a rule base, a database, a decision-making unit, and finally a defuzzification interface as described in Chang(et al 2006). A FIS with five functional block described in Fig.2.

Fig. 2. Fuzzy Inference System Architecture

The function of each block is as follows:

310 Fuzzy Logic – Emerging Technologies and Applications

In this work the humidity and temperature are considered as the greenhouse key

Fuzzy inference systems (FIS) are also known as fuzzy rule-based systems. This is a major unit of a fuzzy logic system. The decision-making is an important part in the entire system. The FIS formulates suitable rules and based upon the rules the decision is made. This is mainly based on the concepts of the fuzzy set theory, fuzzy IF–THEN rules, and fuzzy reasoning. FIS uses "IF - THEN" statements, and the connectors present in the rule

Fuzzy inference system consists of a fuzzification interface, a rule base, a database, a decision-making unit, and finally a defuzzification interface as described in Chang(et al

statement are "OR" or "AND" to make the necessary decision rules.

2006). A FIS with five functional block described in Fig.2.

Fig. 2. Fuzzy Inference System Architecture

parameters, based on (Guzmán-Cruz, *et. Al*, Rico-Garcia, *et al*) observations.

Fig. 1. Greenhouse variable scheme.

**3. Fuzzy systems** 


The working of FIS is as follows. The inputs are converted in to fuzzy by using fuzzification method. After fuzzification the rule base is formed. The rule base and the database are jointly referred to as the knowledge base.

Defuzzification is used to convert fuzzy value to the real world value which is the output.

The steps of fuzzy reasoning (inference operations upon fuzzy IF–THEN rules) performed by FISs are:


#### **4. Fuzzy clustering techniques**

There are a number of fuzzy clustering techniques available. In this work, two fuzzy clustering methods have been chosen: fuzzy c-means clustering and fuzzy clustering subtractive algorithms. These methods are proven to be the most reliable fuzzy clustering methods as well as better forecasters in terms of absolute error according to some authors[Sin, Gomez, Chiu].

Since 1985 when the fuzzy model methodology suggested by Takagi-Sugeno [Takagi *et al*  1985, Sugeno *et al* 1988], as well known as the TSK model, has been widely applied on theoretical analysis, control applications and fuzzy modeling.

Fuzzy system needs the precedent and consequence to express the logical connection between the input output datasets that are used as a basis to produce the desired system behavior [Sin *et al* 1993].

#### **4.1 Fuzzy Clustering Means (FCM)**

Fuzzy C-Means clustering (FCM) is an iterative optimization algorithm that minimizes the cost function given by:

$$J = \sum\_{k=1}^{n} \sum\_{l=1}^{c} \mu\_{lk}^{m} \|\boldsymbol{x}\_{k} - \boldsymbol{v}\_{l}\|^{2} \tag{3}$$

$$\mu\_{\rm lk} = \frac{1}{\sum\_{\parallel=1}^{c} \left( \frac{||\mathbf{x\_k} - \mathbf{v\_l}||}{||\mathbf{x\_k} - \mathbf{v\_l}||} \right)^2 / (\mathbf{m-1})} \tag{4}$$

$$D\_l = \sum\_{l=1}^{n} e^{\left(\frac{\left\|\mathbf{x}\_l - \mathbf{x}\_f\right\|^2}{\left(\frac{r\_a}{2}\right)^2}\right)} \tag{5}$$

$$D\_l = D\_l - D\_{c1}e \left( -\frac{\|\varkappa\_l - \varkappa\_{c1}\|^2}{\left(\frac{\mathcal{T}\_b}{2}\right)^2} \right) \tag{6}$$

$$
\tau\_b = \eta \cdot \tau\_a \tag{7}
$$

$$\mu\_l = e^{-a \| \mathbf{y} \mathbf{y}\_l^\* \|^\ast} \tag{6}$$

$$\alpha = \frac{4}{r\_a^2} \tag{7}$$

$$z = \frac{\sum\_{l=1}^{c} \mu\_l z\_l^\*}{\sum\_{l=1}^{c} \mu\_l} \tag{8}$$

$$\text{IF } \mathbf{x}\_1 \text{ is } A\_1 \text{and } \mathbf{x}\_2 \text{ is } A\_2 \text{and } \dots \text{ THEN } Z\_1 \text{ is } B\_1 \text{and } Z\_2 \text{ is } B\_2 \dots \tag{9}$$

$$A\_l(Q) = \ e^{-a(q - \mathbf{x}\_{lj}^\*)^2} \tag{10}$$

$$B\_{\dagger} = Z\_{\dagger\dagger}^\* \tag{11}$$

$$\mathbf{y}(k) = f(\mathbf{y}(k-n), \mathbf{u}(k)) \tag{12}$$

Greenhouse Fuzzy and Neuro-Fuzzy Modeling Techniques 315

Neurons can combine into a network in numerous fashions. Beyond any doubt the most common of these is the Multilayer Perceptron (MLP) network. The basic MLP-network is constructed by ordering the units in layers, letting each neuron in a layer take as an input only the outputs of neurons in the previous layer or external inputs. Due to the structure, this type of network is often referred to as a feedforward network. [10], [12], [13]. The MLP-network is straightforward to employ for discrete-time modelling of dynamic

**7. The multilayer perceptron** 

Fig. 4. Multilayer perceptron architecture.

fuzzy logic[7][8]. (see figure ).

this node function. In other words, <sup>1</sup>

**8. The Adaptive Neuro-Fuzzy Inference System(ANFIS)** 

**Layer 1:** Every node in i in this layer is a square node with a node function

bell shaped with maximum equal to 1 and minimum equal to 0, such as

In a conventional fuzzy inference system, the number of rules is decided by an expert who is familiar with the system to be modelled. In this particular case study no expert was available and the number of membership functions assigned to each input is chosen empirically. This is carried out by examining the desired input-output data and/or by trial and error. This situation is much the same as ANN's. In this section ANFIS topology and the learning method used for this neuro-fuzzy network are presented. Both neural network and fuzzy logic are model-free estimators and share the common ability to deal with the uncertainties and noise. It is possible to convert fuzzy logic architecture to a neural network and vice versa.[15] This makes it possible to combine the advantages of neural network and

> <sup>1</sup> ( ) *i i 0 Ax*

Where x is the input node i, and *Ai* is the linguistic label(small, large, etc.) associated with

degree to which the *Ai* given x satisfies the quantifier *Ai* . Usually we choose ܣߤሺݔሻ to be

(16)

*<sup>i</sup> 0* is the membership function of and it specifies the

systems.[10]

$$\text{y}(k) = \sum\_{j=1}^{n} a\_j \text{y}(k - j) + \sum\_{j=0}^{n} b\_j u(k - j) + e(k) \tag{13}$$

Te fuzzy system in this case is the proposed by Takagi-Sugeno [8] in which the following equation is presented:

$$\text{IF } \mathbf{x}\_1 \text{ is } A\_1 \text{ and } \mathbf{x}\_2 \text{ is } A\_2 \text{ and } \dots \text{ THEN } \mathbb{Q}(\mathbf{x}) \tag{14}$$

The function *ζ(x)* of the consequence corresponds to a part of a data-cluster as shown on equation 15

$$\mathbf{J}\{\mathbf{x}\} = \mathbf{a}^T \mathbf{x} + \mathbf{b} \tag{15}$$

#### **5. Neural networks**

Artificial neural networks (ANN's) can be used to solve complex problems where noise immunity is important. [12].These feature is why we choose ANN's to model a dynamic system and create a fuzzy inference system. There are two ways to train an ANN: supervised training and un-supervised training. Supervised training requires training set where the input and the desired output of the network are provided for several training cases, whilst un- supervised training requires only the input of the network, and the ANN is supposed to classify (separate) the data appropriately [10]. In this paper we decide to use a supervised ANN because our data source become from experimental measurements.

#### **6. The perceptron**

The neuron or node or unit, as it is called, is a processing element that takes a number of inputs, weights them, sums them up, and uses the result as the argument for a singular valued function, the activation function. (Figure 3) [12], [13].

Fig. 3. Topology of perceptron.

To determine the weight value it is crucial to have a set of samples that correlates the output *yi*, with the, inputs *i*. The task of determining the weights from this example is called training or learning, and is basically a conventional estimation problem.[10]

#### **7. The multilayer perceptron**

314 Fuzzy Logic – Emerging Technologies and Applications

ݕሺ݇ሻ ൌ ܽݕሺ݇ െ ݆ሻ ܾݑሺ݇ െ ݆ሻ ݁ሺ݇ሻ 

Te fuzzy system in this case is the proposed by Takagi-Sugeno [8] in which the following

 *IF x1* is *A1* and *x2* is *A2* and … *THEN ζ(x)* (14) The function *ζ(x)* of the consequence corresponds to a part of a data-cluster as shown on

Artificial neural networks (ANN's) can be used to solve complex problems where noise immunity is important. [12].These feature is why we choose ANN's to model a dynamic system and create a fuzzy inference system. There are two ways to train an ANN: supervised training and un-supervised training. Supervised training requires training set where the input and the desired output of the network are provided for several training cases, whilst un- supervised training requires only the input of the network, and the ANN is supposed to classify (separate) the data appropriately [10]. In this paper we decide to use a

supervised ANN because our data source become from experimental measurements.

valued function, the activation function. (Figure 3) [12], [13].

The neuron or node or unit, as it is called, is a processing element that takes a number of inputs, weights them, sums them up, and uses the result as the argument for a singular

To determine the weight value it is crucial to have a set of samples that correlates the output *yi*, with the, inputs *i*. The task of determining the weights from this example is called

training or learning, and is basically a conventional estimation problem.[10]

(15) ܾݔ்ܽ ൌ ሻݔሺߞ

ୀଵ

equation is presented:

**5. Neural networks** 

**6. The perceptron** 

Fig. 3. Topology of perceptron.

equation 15

ୀ

(13)

Neurons can combine into a network in numerous fashions. Beyond any doubt the most common of these is the Multilayer Perceptron (MLP) network. The basic MLP-network is constructed by ordering the units in layers, letting each neuron in a layer take as an input only the outputs of neurons in the previous layer or external inputs. Due to the structure, this type of network is often referred to as a feedforward network. [10], [12], [13]. The MLP-network is straightforward to employ for discrete-time modelling of dynamic systems.[10]

Fig. 4. Multilayer perceptron architecture.

#### **8. The Adaptive Neuro-Fuzzy Inference System(ANFIS)**

In a conventional fuzzy inference system, the number of rules is decided by an expert who is familiar with the system to be modelled. In this particular case study no expert was available and the number of membership functions assigned to each input is chosen empirically. This is carried out by examining the desired input-output data and/or by trial and error. This situation is much the same as ANN's. In this section ANFIS topology and the learning method used for this neuro-fuzzy network are presented. Both neural network and fuzzy logic are model-free estimators and share the common ability to deal with the uncertainties and noise. It is possible to convert fuzzy logic architecture to a neural network and vice versa.[15] This makes it possible to combine the advantages of neural network and fuzzy logic[7][8]. (see figure ).

**Layer 1:** Every node in i in this layer is a square node with a node function

$$\mathbf{0}\_i^1 = \mu A\_i(\mathbf{x}) \tag{16}$$

Where x is the input node i, and *Ai* is the linguistic label(small, large, etc.) associated with this node function. In other words, <sup>1</sup> *<sup>i</sup> 0* is the membership function of and it specifies the degree to which the *Ai* given x satisfies the quantifier *Ai* . Usually we choose ܣߤሺݔሻ to be bell shaped with maximum equal to 1 and minimum equal to 0, such as

Greenhouse Fuzzy and Neuro-Fuzzy Modeling Techniques 317

Where *wi* is the output of layer 3, and { , , *i ii p q r* } is the parameter set. Parameters in this

**Layer 5:** The single node in this layer is a circle node labelled ∑ that computes the overall

*<sup>w</sup> <sup>f</sup> O overalloutput w f*

Thus we have constructed an adaptive network which is functionally equivalent to a fuzzy inference system [8],[9]. The hybrid algorithm is applied to this architecture. This means that, in the forward pass of the hybrid learning algorithm, functional signals go forward up to fourth layer and the consequent parameters are identified by the least and consequent parameters are identified by the least squares estimation. In the last backward and the

*i i i i*

*i i*

*<sup>w</sup>* (21)

layer will be referred to as consequent parameters.

output as the summation of all incoming signals, ie.

5 1

premise parameters are updated by the gradient descent [8].

Fig. 6. Neural-Netorks Temperature Estimated.

**9. Experimental results** 

$$\mu A\_i(\mathbf{x}) = \frac{1}{1 + \left[ (\frac{\mathbf{x} - \mathbf{c}\_i}{a\_i})^2 \right]^{bi}} \tag{17}$$

where { , , *iii abc* } is the parameter set. As the values of these parameters change, the best bell-shaped functions vary accordingly, thus exhibiting various forms of membership functions on linguistic label *Ai* . In fact, any continuous and piecewise differentiable functions, such as commonly used trapezoidal or triangular-shaped membership functions are also qualified candidates for node functions in this layer. Parameters in this layer are referred to as premise parameters.

Fig. 5. ANFIS Architecture proposed by (Jang 1993).

**Layer 2:** Every node in this layer is a circle node labelled ∏ which multiplies the incoming signals and sends the product out. For instance,

$$
\mu w\_i = \mu A\_i(\mathbf{x}) \* \mu A\_i(y), i = 1, 2 \tag{18}
$$

Each node output represents the firing strength of a rule (In fact, other *T-norm* operators that perform generalized AND can be used as the node function in this layer).

**Layer 3:** Every node in this layer is a circle node labelled N. The ith node calculates the ratio of the ith rule's firing strength to the sum of all rules firing strengths:

$$
\overline{w}\_i = \frac{w\_i}{w\_1 + w\_2}, i = 1, 2. \tag{19}
$$

For convenience, outputs of this layer are called *normalized firing strengths*.

**Layer 4:** Every node in this layer is a square node with a node function

$$^4O\_i^4 = \overline{w}\_i f = \overline{w}(p\_i \mathbf{x} + q\_i \mathbf{y} + r\_i) \tag{20}$$

Where *wi* is the output of layer 3, and { , , *i ii p q r* } is the parameter set. Parameters in this layer will be referred to as consequent parameters.

**Layer 5:** The single node in this layer is a circle node labelled ∑ that computes the overall output as the summation of all incoming signals, ie.

$$MO\_1^5 = overalloutput = \sum\_i \overline{w}\_i f = \frac{\sum\_i w\_i f}{\sum\_i w\_i} \tag{21}$$

Thus we have constructed an adaptive network which is functionally equivalent to a fuzzy inference system [8],[9]. The hybrid algorithm is applied to this architecture. This means that, in the forward pass of the hybrid learning algorithm, functional signals go forward up to fourth layer and the consequent parameters are identified by the least and consequent parameters are identified by the least squares estimation. In the last backward and the premise parameters are updated by the gradient descent [8].

#### **9. Experimental results**

316 Fuzzy Logic – Emerging Technologies and Applications

1( ) *i bi*

where { , , *iii abc* } is the parameter set. As the values of these parameters change, the best bell-shaped functions vary accordingly, thus exhibiting various forms of membership functions on linguistic label *Ai* . In fact, any continuous and piecewise differentiable functions, such as commonly used trapezoidal or triangular-shaped membership functions are also qualified candidates for node functions in this layer. Parameters in this layer are

**Layer 2:** Every node in this layer is a circle node labelled ∏ which multiplies the incoming

( ) ( ), 1,2 *w Ax Ayi ii i*

Each node output represents the firing strength of a rule (In fact, other *T-norm* operators that

**Layer 3:** Every node in this layer is a circle node labelled N. The ith node calculates the ratio

1 2

*<sup>w</sup> w i w w*

, 1,2. *<sup>i</sup>*

(18)

<sup>4</sup> ( ) *O w f w px qy r i i i ii* (20)

(19)

 

perform generalized AND can be used as the node function in this layer).

*i*

For convenience, outputs of this layer are called *normalized firing strengths*.

**Layer 4:** Every node in this layer is a square node with a node function

of the ith rule's firing strength to the sum of all rules firing strengths:

*x c a*

 

<sup>1</sup> ( )

*A x*

referred to as premise parameters.

Fig. 5. ANFIS Architecture proposed by (Jang 1993).

signals and sends the product out. For instance,

2

(17)

*i i*

Fig. 6. Neural-Netorks Temperature Estimated.

Greenhouse Fuzzy and Neuro-Fuzzy Modeling Techniques 319

Fig. 9. ANFIS Humidity Estimated

Fig. 10. Fuzzy Subtractive Clustering Temperature Estimated.

Fig. 7. Neural-Netorks Humidity Estimated

Fig. 8. ANFIS Temperature Estimated.

Fig. 7. Neural-Netorks Humidity Estimated

Fig. 8. ANFIS Temperature Estimated.

Fig. 9. ANFIS Humidity Estimated

Fig. 10. Fuzzy Subtractive Clustering Temperature Estimated.

Greenhouse Fuzzy and Neuro-Fuzzy Modeling Techniques 321

**Mean Average Error Mean Square Error** 

**Algorithm Temperature Humidity Temperature Humidity**  ANN 1.3467 2.8587 4.3418 1.1590 ANFIS 0.3826 1.0634 0.3220 1.0634

Clustering 2.2329 1.7653 12.4100 2.3595 Fuzzy C-Means 1.2329 0.7544 10.4050 1.5533

In this chapter, we have introduced some clustering algorithms for fuzzy model identification, whose main purpose is modeling a system from experimental measured data. Fuzzy model construction by clustering algorithms, however, will need further enhancement. For instance, mechanisms to find values for optimal cluster indexes still need further investigation because, determines the model structure. Here clustering evaluation functions and validation indexes could be of value when combined with genetic algorithms and support vector machines. The effectiveness of this approach will, however, depend on the accuracy of clustering techniques, and the issue still open. These are the questions to be

Fig. 13. Fuzzy C-Means Clustering Humidity Estimated.

Fuzzy Subtractive

Table 1. Summary of Results

addressed in future research.

**10. Conclusions and further work** 

Fig. 11. Fuzzy Subtractive Clustering Humidity Estimated.

Fig. 12. Fuzzy C-Means Clustering Temperature Estimated.

Fig. 11. Fuzzy Subtractive Clustering Humidity Estimated.

Fig. 12. Fuzzy C-Means Clustering Temperature Estimated.

Fig. 13. Fuzzy C-Means Clustering Humidity Estimated.


Table 1. Summary of Results

#### **10. Conclusions and further work**

In this chapter, we have introduced some clustering algorithms for fuzzy model identification, whose main purpose is modeling a system from experimental measured data.

Fuzzy model construction by clustering algorithms, however, will need further enhancement. For instance, mechanisms to find values for optimal cluster indexes still need further investigation because, determines the model structure. Here clustering evaluation functions and validation indexes could be of value when combined with genetic algorithms and support vector machines. The effectiveness of this approach will, however, depend on the accuracy of clustering techniques, and the issue still open. These are the questions to be addressed in future research.

**16** 

 *Iran* 

H. Haroonabadi

**Generation Reliability Evaluation** 

 *Islamic Azad University (IAU)-Dezful Branch,* 

**in Deregulated Power Systems Using** 

**Monte Carlo Simulation and Fuzzy Systems** 

The power systems main emphasis is to provide a reliable and economic supply of electrical energy to the customers (Billinton & Allan, 1996). A real power system is complex, highly integrated and almost very large. It can be divided into appropriate subsystems in order to be analyzed separately (Billinton & Allan, 1996). This research deals with generation reliability assessment in power pool markets. Therefore transmission and distribution

Most of the methods used for generation reliability evaluation are based on the "loss of load or energy" approach. One of the suitable indices that describes generation reliability level is "Loss of Load Expectation" (*LOLE*), that is the time in which load is more than the available

 Reliable Transmission & Distribution Systems

Gen. 2 Load

Generally, the reliability indices of a system can be evaluated using one of the following two

Simulation techniques estimate the reliability indices by simulating the actual process and random behavior of the system. Since power markets and generators' forced outages have stochastic behavior, Monte Carlo Simulation (MCS), as one of the most powerful methods

systems are considered reliable (Hierarchical Levels-I, HL-I) as shown in Fig. 1.

Fig. 1. Power pool market schematic for generation reliability assessment

**1. Introduction** 

generation capacity.

 Analytical techniques Stochastic simulation

basic approaches (Billinton & Allan, 1992):

Gen. 1

Gen. n

#### **11. References**


### **Generation Reliability Evaluation in Deregulated Power Systems Using Monte Carlo Simulation and Fuzzy Systems**

H. Haroonabadi  *Islamic Azad University (IAU)-Dezful Branch, Iran* 

#### **1. Introduction**

322 Fuzzy Logic – Emerging Technologies and Applications

Bezdek, J. C., "Pattern Recognition with Fuzzy Objective Function Algorithms", *Plenum* 

Chang Wook A., "Advances in Evolutionary Algorithms: Theory, Design and Practice",

Chiu S, "Fuzzy model identification based on cluster estimation", *Journal of Intelligent and* 

Ferreira, P.M., E.A. Faria and A.E. Ruano, 2002. Neural network models in greenhouse air

Gomez, A. F., M. Delgado, and M. A. Vila, "About the Use of Fuzzy Clustering Techniques for Fuzzy Model Identification", *Fuzzy Set and System*,. 1999, pp. 179-188. Guzman-Cruz, R., R. Castaneda-Miranda, J.J. Garia-Escalante, I.L. Lopez-Cruz, A. Lara-

Rico-Garcia, E., I.L. Lopez-Cruz, G. Herrera-Ruiz, G.M. Soto-Zarazua and R. Castaneda-

Sugeno, M., and G. T. Kang. "Structure Identification of Fuzzy Model", *Fuzzy Sets and* 

Takagi, T., and M. Sugeno, "Fuzzy Identification of Systems and its Application to Modeling and Control", *IEEE Trans. Systems Man and Cybernetics*. 1985 -*15*, pp. 116-132. Rico-Garcia, E., I.L. Lopez-Cruz, G. Herrera-Ruiz, G.M. Soto-Zarazua and R. Castaneda-

Herrera and J.I. de la Rosa, 2009. *Calibration of a greenhouse climate model using* 

Miranda, 2008. *Effect of temperature on greenhouse natural ventilation under hot conditions: Computational fluid dynamics simulations*. J. Applied Sci., 8: 4543-4551. Rodrigo Castañeda-Miranda; Eusebio Jr. Ventura-Ramos; Rebeca del Rocío Peniche-Vera;

Gilberto Herrera-Ruiz, *Fuzzy Greenhouse Climate Control System based on a Field Programmable Gate Array*, Biosystems Engineering. 2006 Vol. 94/2, pp 165–177 Sin, S. K., and De Figueiredo, "Fuzzy System Designing Through Fuzzy Clustering and

Optimal preDefuzzification", *Proc. IEEE International Conference on Fuzzy Systems*.

Miranda, 2008. *Effect of temperature on greenhouse natural ventilation under hot conditions: Computational fluid dynamics simulations*. J. Applied Sci., 8: 4543-4551. Yager, R. and D. Filev, "Generation of Fuzzy Rules by Mountain Clustering", *Journal of* 

**11. References** 

*Press, NY*, 1981.

1993 *2*, 190-195.

*Systems*. 1988, *28*, pp. 15-33.

Springer, ISSN: 1860-949X, 2006.

*Fuzzy Systems*; September 1994, *2*, pp. 267–78.

temperature prediction. *Neurocomputing,* 2002 43: 51-75

*evolutionary algorithms*. Biosyst. Eng., 104: 135-142

*Intelligent & Fuzzy Systems,* 1994, 2, pp. 209- 219.

The power systems main emphasis is to provide a reliable and economic supply of electrical energy to the customers (Billinton & Allan, 1996). A real power system is complex, highly integrated and almost very large. It can be divided into appropriate subsystems in order to be analyzed separately (Billinton & Allan, 1996). This research deals with generation reliability assessment in power pool markets. Therefore transmission and distribution systems are considered reliable (Hierarchical Levels-I, HL-I) as shown in Fig. 1.

Fig. 1. Power pool market schematic for generation reliability assessment

Most of the methods used for generation reliability evaluation are based on the "loss of load or energy" approach. One of the suitable indices that describes generation reliability level is "Loss of Load Expectation" (*LOLE*), that is the time in which load is more than the available generation capacity.

Generally, the reliability indices of a system can be evaluated using one of the following two basic approaches (Billinton & Allan, 1992):


Simulation techniques estimate the reliability indices by simulating the actual process and random behavior of the system. Since power markets and generators' forced outages have stochastic behavior, Monte Carlo Simulation (MCS), as one of the most powerful methods

Generation Reliability Evaluation in

of demand is explained as:

**2. Power pool markets fundamentals** 

Deregulated Power Systems Using Monte Carlo Simulation and Fuzzy Systems 325

Market demand curve has negative gradient, and the amount of demand decrease is explained by "price elasticity of demand". This index is small for short terms, and big for long terms; because in longer terms, customers can better adjust their load relative to price (IEA, 2003). Demand function, generally, is described as *P=a-b.Q*. Therefore, price elasticity

Let's suppose forecasted load by dispatching center is an independent power from price that

.. . *<sup>n</sup> <sup>n</sup>*

Typically, as shown in Fig. 2, price elasticity in power markets is 0.1-0.2 for the next 2-3

0.15

Offer curve of a company, which participates in a market without any market power is part of the marginal cost curve that is more than minimum average variable cost (Pindyck & Rubinfeld, 1995). Also, total offer curve of all companies is obtained from horizontal sum of each company's offer curve. This curve is a merit order function. In economics, if sale price in a market becomes less than minimum average variable cost, the company will stop production; because the company will not be able to cover not only the fix cost but even the variable cost (Pindyck & Rubinfeld, 1995). Due to the changing efficiency and heat rate of power plants, marginal cost is less than average variable cost. Therefore, in power plants,

average variable cost replaces marginal cost in economic studies (Borenstein, 1999).

0 2.5 15 20 Future time (Year)

*d dQ <sup>E</sup>*

equals to *Qn*. Therefore, demand function can be obtained as:

0.00001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Fig. 2. Price elasticity of demand for various times

 Pric e elas tic ity of demand

years and 0.3-0.7 for the next 10-20 years (IEA, 2003).

1

*d d <sup>Q</sup> <sup>Q</sup> P a bQ bQ bQ E E* (2)

0.5

*dP b* (1)

0.7

for statistical analysis of stochastic problems, is used for reliability assessment in this research.

Generation reliability depends absolutely on the generating units specifications. The main function in traditional structure for Unit Commitment (UC) of the generators is to minimize generation costs. Since the beginning of the 21st century, many countries have been trying to deregulate their power systems and create power markets (Salvaderi, 2000), (Mountford & Austria, 1999), (Draper, 1998), (Puttgen et al, 2001), (Mc Clanahan,2002). In the power markets, the main function of players is their own profit maximization, which severely depends on the type of the market. As a result, generation reliability assessment depends on market type and its characteristics.

Generally, economists divide the markets into four groups, varying between perfect competition market and monopoly market (Pindyck & Rubinfeld, 1995). This study deals with the evaluation of generation reliability in different kinds of power pool markets based on the market concentration. Let's review some of the papers proposed till now.

An optimization technique is proposed in (Wang et al, 2009) to determine load shedding and generation re-dispatch for each contingency state in the reliability evaluation of restructured power systems with the Poolco market structure. The problem is formulated using the optimal power flow (OPF) technique. The objective of the problem is to minimize the total system cost, which includes generation, reserve and interruption costs, subject to market and network constraints.

In reference (Azami, R. et al, 2009) the effect of emergency demand response program on composite system reliability of a deregulated power system is evaluated using an economic load model, AC power-flow-based load curtailment cost function and reliability evaluation techniques.

Reference (Wang & Billinton, 2001) has presented some reliability models for different players in a power system, where generation system is represented by an equivalent multistate generation provider (*EMGP*). The reliability parameters of each *EMGP* are shown by an available capacity probability table (*ACPT*), which is determined using conventional techniques. Then, the equivalent reliability parameters for each state (including state probability, frequency of encountering the state and the equivalent available generation capacity) are determined.

Reference (Haroonabadi & Haghifam, 2009) compares generation reliability in various economic markets: Perfect Competition, Oligopoly and Monopoly power pool markets. Also, due to the stochastic behavior of power market and generators' forced outages, Monte Carlo Simulation is used for reliability evaluation.

In researches dealing with power marketing and restructuring, market behavior and its economic effects on the power system should be considered. Therefore, this research considers power pool market fundamentals and deals with generation reliability assessment in power pool market using MCS and an intelligent system. Also, sensitivity of reliability index to different reserve margins and times will be evaluated. In Section-2, the fundamentals of power pool market will be discussed. In Section-3, the algorithm for generation reliability assessment in power pool market will be proposed, and finally in Section-4, the case study results will be presented and discussed.

#### **2. Power pool markets fundamentals**

324 Fuzzy Logic – Emerging Technologies and Applications

for statistical analysis of stochastic problems, is used for reliability assessment in this

Generation reliability depends absolutely on the generating units specifications. The main function in traditional structure for Unit Commitment (UC) of the generators is to minimize generation costs. Since the beginning of the 21st century, many countries have been trying to deregulate their power systems and create power markets (Salvaderi, 2000), (Mountford & Austria, 1999), (Draper, 1998), (Puttgen et al, 2001), (Mc Clanahan,2002). In the power markets, the main function of players is their own profit maximization, which severely depends on the type of the market. As a result, generation reliability assessment depends on

Generally, economists divide the markets into four groups, varying between perfect competition market and monopoly market (Pindyck & Rubinfeld, 1995). This study deals with the evaluation of generation reliability in different kinds of power pool markets based

An optimization technique is proposed in (Wang et al, 2009) to determine load shedding and generation re-dispatch for each contingency state in the reliability evaluation of restructured power systems with the Poolco market structure. The problem is formulated using the optimal power flow (OPF) technique. The objective of the problem is to minimize the total system cost, which includes generation, reserve and interruption costs, subject to

In reference (Azami, R. et al, 2009) the effect of emergency demand response program on composite system reliability of a deregulated power system is evaluated using an economic load model, AC power-flow-based load curtailment cost function and reliability evaluation

Reference (Wang & Billinton, 2001) has presented some reliability models for different players in a power system, where generation system is represented by an equivalent multistate generation provider (*EMGP*). The reliability parameters of each *EMGP* are shown by an available capacity probability table (*ACPT*), which is determined using conventional techniques. Then, the equivalent reliability parameters for each state (including state probability, frequency of encountering the state and the equivalent available generation

Reference (Haroonabadi & Haghifam, 2009) compares generation reliability in various economic markets: Perfect Competition, Oligopoly and Monopoly power pool markets. Also, due to the stochastic behavior of power market and generators' forced outages, Monte

In researches dealing with power marketing and restructuring, market behavior and its economic effects on the power system should be considered. Therefore, this research considers power pool market fundamentals and deals with generation reliability assessment in power pool market using MCS and an intelligent system. Also, sensitivity of reliability index to different reserve margins and times will be evaluated. In Section-2, the fundamentals of power pool market will be discussed. In Section-3, the algorithm for generation reliability assessment in power pool market will be proposed, and finally in

on the market concentration. Let's review some of the papers proposed till now.

research.

market type and its characteristics.

market and network constraints.

capacity) are determined.

Carlo Simulation is used for reliability evaluation.

Section-4, the case study results will be presented and discussed.

techniques.

Market demand curve has negative gradient, and the amount of demand decrease is explained by "price elasticity of demand". This index is small for short terms, and big for long terms; because in longer terms, customers can better adjust their load relative to price (IEA, 2003). Demand function, generally, is described as *P=a-b.Q*. Therefore, price elasticity of demand is explained as:

$$E\_d = \left| \frac{dQ}{dP} \right| = \frac{1}{b} \tag{1}$$

Let's suppose forecasted load by dispatching center is an independent power from price that equals to *Qn*. Therefore, demand function can be obtained as:

$$P = a - b.Q = b.Q\_n - b.Q = \frac{Q\_n}{E\_d} - \frac{Q}{E\_d} \tag{2}$$

Typically, as shown in Fig. 2, price elasticity in power markets is 0.1-0.2 for the next 2-3 years and 0.3-0.7 for the next 10-20 years (IEA, 2003).

Fig. 2. Price elasticity of demand for various times

Offer curve of a company, which participates in a market without any market power is part of the marginal cost curve that is more than minimum average variable cost (Pindyck & Rubinfeld, 1995). Also, total offer curve of all companies is obtained from horizontal sum of each company's offer curve. This curve is a merit order function. In economics, if sale price in a market becomes less than minimum average variable cost, the company will stop production; because the company will not be able to cover not only the fix cost but even the variable cost (Pindyck & Rubinfeld, 1995). Due to the changing efficiency and heat rate of power plants, marginal cost is less than average variable cost. Therefore, in power plants, average variable cost replaces marginal cost in economic studies (Borenstein, 1999).

Generation Reliability Evaluation in

concentrated (FTC, 1992).

concentrated markets' fuzzy sets

margin, which is defined as (IEA, 2002):

pool markets.

Deregulated Power Systems Using Monte Carlo Simulation and Fuzzy Systems 327

If market shares are measured in percentages, *HHI* will vary between 0 (an atomistic market) and 10000 (monopoly market). According to a usual grouping, the US merger guidelines stipulate an assumption that markets with a *HHI* below 1000 is unconcentrated, a *HHI* between 1000 and 1800 is moderately concentrated, and a *HHI* above 1800 is highly

As mentioned before, according to the type of market and *HHI* values, negative gradient of demand exponent curve varies between *b* and *2b*. Therefore, for modeling the market, a fuzzy number is proposed in this study to estimate the gradient coefficient of demand exponent curve (*K*) based on the *HHI* values. Membership functions of unconcentrated, moderately concentrated and highly concentrated markets' fuzzy sets and the equation to

Fig. 4. Membership functions of unconcentrated, moderately concentrated and highly

As Fig. 4 and (7) show, while the proposed coefficient (*K*) covers all kinds of markets with different concentration degrees, the changes of these degrees are not sudden, rather they are gradual and continuous. Also, the proposed method and fuzzy logic are valid for all power

Generation reliability of a power system depends on many parameters, especially on reserve

% <sup>100</sup> *Installed Capacity Peak Demand RM Peak Demand*

*K MFU MFM MFH* ( 1.5 2 ) (7)

(8)

estimate gradient coefficient are shown in Fig. 4 and (7), respectively.

2 *i M*

*HHI q* (6)

In a perfect competition market, equilibrium price and equilibrium amount are obtained from the intersection of total offer curve and demand curve. On the other hand, in a monopoly market, the monopolist considers the production level, which maximizes his profit. It is proved that the monopolist considers the level of production in which marginal cost of each firm (and total marginal cost of all firms) equals to the marginal revenue of the monopolist (Pindyck & Rubinfeld, 1995):

$$\text{MCC}\_1 = \text{MC}\_2 = \dots = \text{MC} = \text{MR} \tag{3}$$

Where:

$$MR = a - 2.b.\\ Q = b.Q\_n - 2.b.Q = \frac{Q\_n}{E\_d} - \frac{2.Q}{E\_d} \tag{4}$$

Comparison of (2) and (4) shows that if there is no market power, offer curve of industry for each market (from perfect competition market to monopoly market) will equal marginal cost; but negative gradient of demand exponent curve (*DE*) varies between *b* (for demand function in perfect competition market) and *2b* (for marginal revenue in monopoly market). Therefore, generally, demand exponent curve can be expressed as:

$$DE = a - K.b.Q = \frac{Q\_n}{E\_d} - \frac{K.Q}{E\_d} \tag{5}$$

Where, *K* varies between 1 and 2.

Fig. 3 shows the typical total offer and demand exponent curves.

Fig. 3. Typical total offer and demand exponent curves

#### **3. Proposed method for generation evaluation in power markets**

In power markets, Hirschman-Herfindahl Index (*HHI*), which is obtained from (6), is used for market concentration measurement (IEA, 2003):

In a perfect competition market, equilibrium price and equilibrium amount are obtained from the intersection of total offer curve and demand curve. On the other hand, in a monopoly market, the monopolist considers the production level, which maximizes his profit. It is proved that the monopolist considers the level of production in which marginal cost of each firm (and total marginal cost of all firms) equals to the marginal revenue of the

2. 2. . . 2. . *<sup>n</sup> <sup>n</sup>*

. . . *<sup>n</sup>*

*d d*

Comparison of (2) and (4) shows that if there is no market power, offer curve of industry for each market (from perfect competition market to monopoly market) will equal marginal cost; but negative gradient of demand exponent curve (*DE*) varies between *b* (for demand function in perfect competition market) and *2b* (for marginal revenue in monopoly market).

Therefore, generally, demand exponent curve can be expressed as:

Fig. 3 shows the typical total offer and demand exponent curves.

Fig. 3. Typical total offer and demand exponent curves

for market concentration measurement (IEA, 2003):

**3. Proposed method for generation evaluation in power markets** 

In power markets, Hirschman-Herfindahl Index (*HHI*), which is obtained from (6), is used

1 2 *MC MC MC MR* ... (3)

*d d <sup>Q</sup> <sup>Q</sup> MR a b Q b Q b Q E E* (4)

*<sup>Q</sup> K Q DE a K b Q E E* (5)

monopolist (Pindyck & Rubinfeld, 1995):

Where, *K* varies between 1 and 2.

Where:

$$HHI = \sum\_{M} q\_i^2 \tag{6}$$

If market shares are measured in percentages, *HHI* will vary between 0 (an atomistic market) and 10000 (monopoly market). According to a usual grouping, the US merger guidelines stipulate an assumption that markets with a *HHI* below 1000 is unconcentrated, a *HHI* between 1000 and 1800 is moderately concentrated, and a *HHI* above 1800 is highly concentrated (FTC, 1992).

As mentioned before, according to the type of market and *HHI* values, negative gradient of demand exponent curve varies between *b* and *2b*. Therefore, for modeling the market, a fuzzy number is proposed in this study to estimate the gradient coefficient of demand exponent curve (*K*) based on the *HHI* values. Membership functions of unconcentrated, moderately concentrated and highly concentrated markets' fuzzy sets and the equation to estimate gradient coefficient are shown in Fig. 4 and (7), respectively.

Fig. 4. Membership functions of unconcentrated, moderately concentrated and highly concentrated markets' fuzzy sets

$$K = \left(M\text{FLI} + \mathbf{1.5} \times M\text{FMI} + \mathbf{2} \times M\text{FHI}\right) \tag{7}$$

As Fig. 4 and (7) show, while the proposed coefficient (*K*) covers all kinds of markets with different concentration degrees, the changes of these degrees are not sudden, rather they are gradual and continuous. Also, the proposed method and fuzzy logic are valid for all power pool markets.

Generation reliability of a power system depends on many parameters, especially on reserve margin, which is defined as (IEA, 2002):

$$\text{RM\%} = \frac{\text{Installed} \quad \text{Capacity} - \text{Peak} \quad \text{Demand}}{\text{Peak} \quad \text{Demand}} \times 100 \tag{8}$$

Generation Reliability Evaluation in

using MCS

**4. Numerical studies** 

used in various case studies:

Deregulated Power Systems Using Monte Carlo Simulation and Fuzzy Systems 329

Fig. 6. Algorithm of available generated power and *LOLE* calculations for each iteration

IEEE - Reliability Test System (IEEE-RTS) is used for case studies. The required data for IEEE-RTS can be found in (Reliability Test System…, 1979). The following assumptions are

6. The steps 3 to 5 are repeated for calculation of final *LOLE*.

for all power plants using an independent random number generated for each plant. Finally, sum of the available power plants' generation capacities is calculated. If the sum becomes less than the intersection of power plants' total offer curve and demand exponent curve, we will have interruption in the iteration, and therefore, *LOLE* will increase one unit; otherwise, we will go to the next iteration. The algorithm of available generated power and *LOLE* calculations for each iteration in MCS is shown in Fig. 6.

The algorithm of generation reliability assessment in power pool markets using Monte Carlo simulation and proposed fuzzy logic is as follows (Fig. 5):

Fig. 5. Flow chart of HLI reliability assessment in power markets using MCS


The algorithm of generation reliability assessment in power pool markets using Monte Carlo

Fig. 5. Flow chart of HLI reliability assessment in power markets using MCS

demand exponent curve (*K*) is calculated using Fig. 4 and (7).

2. Calculation of the total offer curve of power plants.

regards to the reserve margin.

1. *HHI* is obtained based on characteristic of the market. The gradient coefficient of

3. Select a random day and its load (*Qn*), and calculate demand exponent curve using (5). 4. The power plants, selected for generation in the selected day, are determined from the intersection of the power plants' total offer curve and demand exponent curve with

5. For each selected power plant in the previous step, a random number between 0-1 is generated. If the generated number is more than the power plant's Forced Outage Rate (*FOR*), the power plant is considered as available in the mentioned iteration; otherwise it encounters forced outage and thus can not generate power. This process is performed

simulation and proposed fuzzy logic is as follows (Fig. 5):

for all power plants using an independent random number generated for each plant. Finally, sum of the available power plants' generation capacities is calculated. If the sum becomes less than the intersection of power plants' total offer curve and demand exponent curve, we will have interruption in the iteration, and therefore, *LOLE* will increase one unit; otherwise, we will go to the next iteration. The algorithm of available generated power and *LOLE* calculations for each iteration in MCS is shown in Fig. 6.

6. The steps 3 to 5 are repeated for calculation of final *LOLE*.

Fig. 6. Algorithm of available generated power and *LOLE* calculations for each iteration using MCS

#### **4. Numerical studies**

IEEE - Reliability Test System (IEEE-RTS) is used for case studies. The required data for IEEE-RTS can be found in (Reliability Test System…, 1979). The following assumptions are used in various case studies:

Generation Reliability Evaluation in

the second study

Deregulated Power Systems Using Monte Carlo Simulation and Fuzzy Systems 331

In the second study, all the power plants based on their types (including oil, coal, nuclear and water plants) are classified. Therefore, *HHI* equals 2984, and *K* is calculated as 1.5722 (Fig. 9). Based on this assumption and using MCS algorithm, *LOLE* values are obtained

Fig. 9. The gradient calculation of demand exponent curve using membership functions for

**HHI=2984 & K=1.5722**

RM=0% 50.96 10.26 RM=9% 36.95 5.46

In the third study, all fossil power plants (including oil and coal power plants) are classified in one company, and other power plants are as in the second case study. Therefore, the types of power plants are fossil, nuclear and water. As a result, *HHI* equals 5290, and *K* is calculated as 1.7128 (Fig. 11). Based on this assumption and using MCS algorithm, *LOLE*

values are obtained versus different times and reserve margins as shown in Fig. 12.

Future years

0 2

versus different times and reserve margins as shown in Fig. 10.

LOLE

Fig. 10. *LOLE* values for the second study

(days / second half of year)


In the first case study, each power plant is assumed as an independent company. Therefore, *HHI* equals 634, and the market is unconcentrated. Using Fig. 4 and (7), *K* is calculated as 1 (Fig. 7). Based on this assumption and using MCS algorithm, *LOLE* values are obtained versus different times and reserve margins as shown in Fig. 8.

Fig. 7. The gradient calculation of demand exponent curve using membership functions for the first study

**HHI=634 & K=1**

Fig. 8. *LOLE* values for the first study

1. All case studies are simulated for the second half of the year, based on the daily peak

3. Each case study is simulated for two different times (present time and the 2nd next year)

4. Annual growth rates of the power plants' generation capacity and consumed load are

5. Annual growth rates of oil and coal costs are considered as 4% and 1%, respectively. Nuclear fuel cost (including uranium, enrichment and fabrication) is considered as a fixed rate. Also, annual growth rate of variable Operating and Maintenance (O&M) cost

In the first case study, each power plant is assumed as an independent company. Therefore, *HHI* equals 634, and the market is unconcentrated. Using Fig. 4 and (7), *K* is calculated as 1 (Fig. 7). Based on this assumption and using MCS algorithm, *LOLE* values are obtained

Fig. 7. The gradient calculation of demand exponent curve using membership functions for

**HHI=634 & K=1**

RM=0% 69.52 12.7 RM=9% 42.19 11.47

Future years

0 2

load of the mentioned test system. 2. All simulations are done with 5000 iterations.

is considered as 1%.

the first study

and two different reserve margins (0%, 9%).

considered as 3.4% and 3.34%, respectively.

versus different times and reserve margins as shown in Fig. 8.

LOLE

Fig. 8. *LOLE* values for the first study

(days / second half of year)

In the second study, all the power plants based on their types (including oil, coal, nuclear and water plants) are classified. Therefore, *HHI* equals 2984, and *K* is calculated as 1.5722 (Fig. 9). Based on this assumption and using MCS algorithm, *LOLE* values are obtained versus different times and reserve margins as shown in Fig. 10.

Fig. 9. The gradient calculation of demand exponent curve using membership functions for the second study

**HHI=2984 & K=1.5722**

Fig. 10. *LOLE* values for the second study

In the third study, all fossil power plants (including oil and coal power plants) are classified in one company, and other power plants are as in the second case study. Therefore, the types of power plants are fossil, nuclear and water. As a result, *HHI* equals 5290, and *K* is calculated as 1.7128 (Fig. 11). Based on this assumption and using MCS algorithm, *LOLE* values are obtained versus different times and reserve margins as shown in Fig. 12.

Generation Reliability Evaluation in

the fourth study

improve.

decrease.

Deregulated Power Systems Using Monte Carlo Simulation and Fuzzy Systems 333

Fig. 13. The gradient calculation of demand exponent curve using membership functions for

**HHI=10000 & K=2**

RM=0% 32.72 10.03 RM=9% 19.22 5.27

In all case studies, if reserve margin increases, *LOLE* will decrease and reliability will

As mentioned before, in longer terms, customers can better adjust their load relative to the price. Therefore, price elasticity increases in longer terms, and according to (5), demand exponent curve reaches less gradient. As a result, intersection of the power plants' total offer curve and demand exponent curve will occur at less demand. This matter leads to operate from fewer power plants. Therefore, in each case study, if time increases, *LOLE* will

Future years

0 2

LOLE

Fig. 14. *LOLE* values for the fourth study

(days / second half of year)

Fig. 11. The gradient calculation of demand exponent curve using membership functions for the third study

**HHI=5290 & K=1.7128**

Fig. 12. *LOLE* values for the third study

In the fourth study, it is assumed that all power plants belong to a monopolist, and the market is fully concentrated and monopoly. Therefore, *HHI* equals 10000, and *K* is calculated as 2 (Fig. 13). Based on this assumption and using MCS algorithm, *LOLE* values are obtained versus different times and reserve margins as shown in Fig. 14.

Fig. 11. The gradient calculation of demand exponent curve using membership functions for

**HHI=5290 & K=1.7128**

RM=0% 41.02 10.14 RM=9% 35.74 5.38

In the fourth study, it is assumed that all power plants belong to a monopolist, and the market is fully concentrated and monopoly. Therefore, *HHI* equals 10000, and *K* is calculated as 2 (Fig. 13). Based on this assumption and using MCS algorithm, *LOLE* values

are obtained versus different times and reserve margins as shown in Fig. 14.

Future years

0 2

LOLE

Fig. 12. *LOLE* values for the third study

(days / second half of year)

the third study

Fig. 13. The gradient calculation of demand exponent curve using membership functions for the fourth study

#### **HHI=10000 & K=2**

Fig. 14. *LOLE* values for the fourth study

In all case studies, if reserve margin increases, *LOLE* will decrease and reliability will improve.

As mentioned before, in longer terms, customers can better adjust their load relative to the price. Therefore, price elasticity increases in longer terms, and according to (5), demand exponent curve reaches less gradient. As a result, intersection of the power plants' total offer curve and demand exponent curve will occur at less demand. This matter leads to operate from fewer power plants. Therefore, in each case study, if time increases, *LOLE* will decrease.

Generation Reliability Evaluation in

*HHI*: Hirschman - Herfindahl index *DE*: Demand exponent curve

*AGP*: Available generated power

*Conference*, pp. 1-8.

**7. References** 

2001.

*K*: Gradient coefficient of demand exponent curve *MFU*: Membership function of unconcentrated market

*MFM*: Membership function of moderately concentrated market *MFH*: Membership function of highly concentrated market *NG*: Number of selected plants for generation in the market

Plenum press, ISBN: 0-306-44063-6, New York.

press, ISBN: 0-306-45259-6, New York.

Deregulated Power Systems Using Monte Carlo Simulation and Fuzzy Systems 335

Azami R., Abbasi A.H., Shakeri J., Fard A.F. (2009), Impact of EDRP on Composite

Billinton R., Allan R. (1992). *Reliability Evaluation of Engineering Systems*, Second edition,

Billinton R., Allan R. (1996). *Reliability Evaluation of Power Systems*, Second edition, Plenum

Borenstein Serverin (1999). Understanding competitive pricing and market power in

Draper E. L. (1998). Assessment of Deregulation and Competition, *IEEE Power Engineering* 

Haroonabadi H. & Haghifam M.-R. (2009). Generation Reliability Evaluation in Power

International Energy Agency (IEA) (2002). *Security of Supply in Electricity Markets - Evidence* 

International Energy Agency (IEA) (2003). *The Power to Choose- Demand Response in* 

Jaeseok Choi; Hongsik Kim; Junmin Cha & Roy Billinton (2001). Nodal probabilistic

Mc Clanahan R. H. (2002). Electric Deregulation, *IEEE Industry Application Magazine*, Vol. 8,

Mountford J. D., Austria R. R. (1999). Keeping The Lights On, *IEEE Spectrum*, Vol. 36 (Jun

Pindyck Robert S. & Rubinfeld D. L. (1995). *Microeconomics*, Third edition, Prentice Hall,

Puttgen H. B.; Volzka D. R. & Olken M. I. (2001). Restructuring and Reregulation of The US

Reliability Test System Task Force of The IEEE Subcommittee on the application of

Electric Utility Industry*, IEEE Power Engineering Review*, Vol. 21, No. 2 (Feb 2001),

probability Methods, IEEE Reliability Test System, *IEEE Transactions*, Pas-98, No.6,

Markets Using Monte Carlo Simulation and Neural Networks*. Proceedings of 15th International Conference on Intelligent System Applications to Power Systems (ISAP)*, pp.

congestion and reliability evaluations of a transmission system under the deregulated electricity market, *Proceedings of IEEE Power engineering society summer meeting,,* pp. 497-502, Print ISBN: 0-7803-7173-9, Vancouver, 15 Jul 2001-19 Jul

wholesale electricity market, *University of California energy institute*.

*Review*, Vol. 18, No. 7 (Jul 1998), pp. 17-18, ISSN: 0272-1724.

*Liberalized Electricity Markets,* IEA, ISBN: 92-64-10503-4, France.

1-6, Print ISBN: 978-1-4244-5097-8, Curitiba, Nov 2009.

*and Policy Issues*, IEA, ISBN: 92-64-19805-9, France.

No. 2 (Mar/Apr 2002), pp. 11-18, ISSN: 1077-2618 .

1999), pp. 34-39, ISSN: 0018-9235.

ISBN: 7-302-02494-4, USA.

pp. 8-10, ISSN: 0272-1724.

Nov/Dec 1979, pp. 2047-2054.

Reliability of Restructured Power Systems, *Proceedings of PowerTech IEEE Bucharest* 

If market becomes more concentrated or *HHI* becomes bigger, *K* will find bigger value. Therefore, according to (5), intersection of the power plants' total offer curve and demand exponent curve will occur at less demand. Therefore, *LOLE* will decrease. So that in the fourth study (monopoly market), *LOLE*s are the least values comparing to the other case studies.

It is to be noted that since available capacity of hydro plants in IEEE-RTS are different in the first and the second halves of the year, therefore, simulations were done for the second half of the year. Evidently, the proposed method can be utilized for every simulation time. Also, in this study, it was supposed that the annual additional generation capacity is uniformly distributed between all the present generators.

### **5. Conclusion**

This research deals with generation reliability assessment in power pool market using Monte Carlo simulation and intelligent systems. Since changes of market concentration in power markets are gradual, a fuzzy logic was proposed for calculation of the gradient coefficient of demand exponent curve. Due to the stochastic behavior of market and generators' *FOR*, MCS was used for the simulations. In this research, *LOLE* was used as reliability index and it was shown that if market becomes more concentrated, *LOLE* will decrease and reliability will improve. Also, if price elasticity of demand increases, *LOLE* will decrease.

Follows can be considered for future researches:


#### **6. Symbol List**

*MC*: Marginal cost (mills/kWh) *MR*: Marginal revenue (mills/kWh) *5Q*: Quantity of power (kW) *P*: Electrical energy price (mills/kWh) *RM*: Reserve margin (%) *Ed* : Price elasticity of demand (kW2h/mills) *Qn*: Forecasted load (kW) *LOLE*: Loss of load expectation (days/second half year) *FOR*: Forced outage rate of power plants *qi*: Share of ith company in the pool market (%) *M*: Number of independent companies in the market *a*: Demand exponent curve cross of basis (mills/kWh) *b*: Demand exponent curve gradient (mills /kW2h)


*AGP*: Available generated power

#### **7. References**

334 Fuzzy Logic – Emerging Technologies and Applications

If market becomes more concentrated or *HHI* becomes bigger, *K* will find bigger value. Therefore, according to (5), intersection of the power plants' total offer curve and demand exponent curve will occur at less demand. Therefore, *LOLE* will decrease. So that in the fourth study (monopoly market), *LOLE*s are the least values comparing to the other case

It is to be noted that since available capacity of hydro plants in IEEE-RTS are different in the first and the second halves of the year, therefore, simulations were done for the second half of the year. Evidently, the proposed method can be utilized for every simulation time. Also, in this study, it was supposed that the annual additional generation capacity is uniformly

This research deals with generation reliability assessment in power pool market using Monte Carlo simulation and intelligent systems. Since changes of market concentration in power markets are gradual, a fuzzy logic was proposed for calculation of the gradient coefficient of demand exponent curve. Due to the stochastic behavior of market and generators' *FOR*, MCS was used for the simulations. In this research, *LOLE* was used as reliability index and it was shown that if market becomes more concentrated, *LOLE* will decrease and reliability will improve. Also, if price elasticity of demand increases, *LOLE* will

1. Reliability indices evaluate in HL-II zone in which both generation and transmission

3. If the generation planning scenarios in a power system are specified, then they can be used instead of uniformly distribution of annual additional generation capacity. 4. Reserve market can be considered as an independent market of the main energy

2. Bilateral contracts consider in the power market as well as pool market.

studies.

**5. Conclusion** 

decrease.

market.

**6. Symbol List** 

distributed between all the present generators.

Follows can be considered for future researches:

systems are considered.

*MC*: Marginal cost (mills/kWh) *MR*: Marginal revenue (mills/kWh)

*P*: Electrical energy price (mills/kWh)

*Ed* : Price elasticity of demand (kW2h/mills)

*FOR*: Forced outage rate of power plants *qi*: Share of ith company in the pool market (%) *M*: Number of independent companies in the market *a*: Demand exponent curve cross of basis (mills/kWh) *b*: Demand exponent curve gradient (mills /kW2h)

*LOLE*: Loss of load expectation (days/second half year)

*5Q*: Quantity of power (kW)

*RM*: Reserve margin (%)

*Qn*: Forecasted load (kW)


Salvaderi L. (2000). Electric Sector Restructuring in Italy, *IEEE Power Engineering Review*, Vol.

Wang P. & Billinton R. (2001). Implementation of non-uniform reliability in a deregulated

*Engineerin,.* pp. 857- 861, Print ISBN: 0-7803-6715-4, Toronto, May 2001. Wang P.; Ding, Y. & Goel, L. (2009). Reliability assessment of restructured power systems

power market, *Proceedings of Canadian Conference on Electrical and Computer* 

using optimal load shedding technique, *Generation, Transmission & Distribution, IET*,

20, No. 4 (Apr 2000), pp. 12-16, ISSN: 0272-1724.

http://www.ftc.gov/bc/docs/horizmer.htm.

The U.S. Department of Justice and Federal Trade Commission (FTC) (1992).

Vol. 3, Issue: 7 (July 2009), pp. 628 – 640, ISSN: 1751-8687.

### *Edited by Elmer P. Dadios*

The capability of Fuzzy Logic in the development of emerging technologies is introduced in this book. The book consists of sixteen chapters showing various applications in the field of Bioinformatics, Health, Security, Communications, Transportations, Financial Management, Energy and Environment Systems. This book is a major reference source for all those concerned with applied intelligent systems. The intended readers are researchers, engineers, medical practitioners, and graduate students interested in fuzzy logic systems.

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Fuzzy Logic - Emerging Technologies and Applications

Fuzzy Logic

Emerging Technologies and Applications